Enhanced Binding in Quantum FieldTheory
Fumio Hiroshima
Faculty of Mathematics, Kyushu UniversityFukuoka, Japan
Itaru Sasaki
Department of Engineering, Shinshu UniversityMatsumoto, Japan
Herbert Spohn
Mathematical Zentrum, Technische Universitat MunchenMunchen, Germany
Akito Suzuki
Department of Engineering, Shinshu UniversityNagano, Japan
October 17, 2018arX
iv:1
203.
1136
v1 [
mat
h-ph
] 6
Mar
201
2
Abstract
Enhanced binding in quantum field theory and related topics are reviewed, whichsuggests new possibility, beyond toy models, in the study of the stability of quantumfield models. This Lecture Note reviews papers below:
1. F. Hiroshima and H. Spohn, Enhanced binding through coupling to quantum field,Ann. Henri Poincare 2 (2001), 1159–1187.
2. F. Hiroshima and I. Sasaki, Enhanced binding of an N particle system interacting witha scalar field I, Math. Z. 259 (2008), 657–680.
3. F. Hiroshima, H. Spohn and A. Suzuki, The no-binding regime of the Pauli-Fierz model,J. Math. Phys. 52 (2011), 062104.
4. C. Gerard, F. Hiroshima, A. Panati, and A. Suzuki, Absence of ground state of theNelson model with variable coefficients, J. Funct. Anal. 262 (2012), 273–299.
This lecture note consists of three parts. Fundamental facts on Boson Fock space areintroduced in Part I. Ref. 1.and 3. are reviewed in Part II and, Ref. 2. and 4. in Part III.
In Part I a symplectic structure of a Boson Fock space is studied and a projectiveunitary representation of an infinite dimensional symplectic group through Bogoliubovtransformations is constructed.
In Part II the so-called Pauli-Fierz model (PF model) with the dipole approxima-tion in non-relativistic quantum electrodynamics is investigated. This model describes aminimal interaction between a massless quantized radiation field and a quantum mechan-ical particle (electron) governed by Schrodinger operator. By applying the Bogoliubovtransformation introduced in Part I we investigate the spectrum of the PF model. Firstthe translation invariant case is considered and the dressed electron state with a fixedmomentum is studied. Secondly the absence of ground state is proven by extending theBirman-Schwinger principle. Finally the enhanced binding of a ground state is discussedand the transition from unbinding to binding is shown.
In Part III the so-called N -body Nelson model is studied. This model describes alinear interaction between a scalar field and N -body quantum mechanical particles. Firstthe enhanced binding is shown by checking the so-called stability condition. Secondly theNelson model with variable coefficients is discussed, which model can be derived when theMinkowskian space-time is replaced by a static Riemannian manifold, and the absence ofground state is proven, if the variable mass decays to zero sufficiently fast. The strategyis based on a path measure argument.
Contents
I Boson Fock space and symplectic structures 5
1 Boson Fock space 51.1 Second quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2 Segal fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.3 Wick product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2 Symplectic structure 132.1 Infinite dimensional symplectic group . . . . . . . . . . . . . . . . . . 132.2 Quadratic operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.3 Bogoliubov transformations . . . . . . . . . . . . . . . . . . . . . . . 20
2.3.1 Homogeneous case . . . . . . . . . . . . . . . . . . . . . . . . 202.3.2 Inhomogeneous case . . . . . . . . . . . . . . . . . . . . . . . 22
2.4 One parameter symplectic groups and 2-cocycles . . . . . . . . . . . . 25
II The Pauli-Fierz model 29
3 The Pauli-Fierz Hamiltonian 293.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.2 The Pauli-Fierz Hamiltonian with the dipole approximation . . . . . 313.3 Translation invariant Hamiltonian . . . . . . . . . . . . . . . . . . . . 343.4 Bogoliubov transformation . . . . . . . . . . . . . . . . . . . . . . . . 38
3.4.1 Algebraic relations . . . . . . . . . . . . . . . . . . . . . . . . 383.4.2 Intertwining operator . . . . . . . . . . . . . . . . . . . . . . . 533.4.3 Displacement operator . . . . . . . . . . . . . . . . . . . . . . 57
3.5 Diagonalization and time evolution of radiation fields . . . . . . . . . 593.5.1 Diagonalization . . . . . . . . . . . . . . . . . . . . . . . . . . 593.5.2 Time evolution of quantized radiation field . . . . . . . . . . . 63
3.6 Dressed electron states . . . . . . . . . . . . . . . . . . . . . . . . . . 643.7 Ground state energy . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.7.1 Holomorphic property . . . . . . . . . . . . . . . . . . . . . . 663.7.2 Explicit form of effective mass and ground state energy . . . . 713.7.3 Ultraviolet cutoffs . . . . . . . . . . . . . . . . . . . . . . . . . 793.7.4 Many particle system . . . . . . . . . . . . . . . . . . . . . . . 83
2
3.8 Self-energy term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 843.8.1 Diagonalization and DES . . . . . . . . . . . . . . . . . . . . . 843.8.2 No self-enery term . . . . . . . . . . . . . . . . . . . . . . . . 87
3.9 Scaling limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 883.9.1 Weak coupling limit . . . . . . . . . . . . . . . . . . . . . . . 883.9.2 Strong coupling limit . . . . . . . . . . . . . . . . . . . . . . . 90
3.10 Negative mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
4 Binding and non-binding 994.1 Enhanced binding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 994.2 Absence of ground state . . . . . . . . . . . . . . . . . . . . . . . . . 1014.3 Existence of ground state . . . . . . . . . . . . . . . . . . . . . . . . . 107
4.3.1 Massive case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1094.3.2 Massless case . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
4.4 Transition from unbinding to binding . . . . . . . . . . . . . . . . . . 1204.5 Enhanced binding by UV cutoff . . . . . . . . . . . . . . . . . . . . . 121
III The Nelson model 123
5 The Nelson Hamiltonian 1235.1 The Nelson Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . 1235.2 Enhanced binding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1245.3 Weak coupling limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
6 Binding 1296.1 Existence of ground states . . . . . . . . . . . . . . . . . . . . . . . . 1296.2 Stability conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1306.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
6.3.1 Example of effective potential . . . . . . . . . . . . . . . . . . 1376.3.2 Example of external potential . . . . . . . . . . . . . . . . . . 138
7 Absence of ground state 1497.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
7.1.1 Stability and the decay of variable mass . . . . . . . . . . . . 1497.1.2 Klein-Gordon equation on pseudo Riemannian manifold . . . . 150
7.2 The Nelson model with a variable mass . . . . . . . . . . . . . . . . . 1557.2.1 Dirichlet forms and symmetric semigroups . . . . . . . . . . . 155
3
7.2.2 Schrodinger operators with divergence form . . . . . . . . . . 1577.2.3 Scalar quantum fields . . . . . . . . . . . . . . . . . . . . . . . 1617.2.4 The Nelson model with a variable mass . . . . . . . . . . . . . 162
7.3 Feynman-Kac formula and diffusions . . . . . . . . . . . . . . . . . . 1657.3.1 Super-exponential decay . . . . . . . . . . . . . . . . . . . . . 1657.3.2 Diffusion processes . . . . . . . . . . . . . . . . . . . . . . . . 174
7.4 Absence of ground state . . . . . . . . . . . . . . . . . . . . . . . . . 1827.4.1 The Nelson model by path measures . . . . . . . . . . . . . . 1827.4.2 Absence of ground states . . . . . . . . . . . . . . . . . . . . . 184
4
Part I
Boson Fock space and symplecticstructures
1 Boson Fock space
1.1 Second quantization
Let h be a separable Hilbert space over the complex field C with the scalar product
(·, ·)h. Here the scalar product is linear in the second component and antilinear in the
first one. We omit subscript h unless confusion may arise. Consider the operation
⊗ns of n-fold symmetric tensor product defined through the symmetrization operator
Sn(f1 ⊗ · · · ⊗ fn) =1
n!
∑π∈℘n
fπ(1) ⊗ · · · ⊗ fπ(n), n ≥ 1, (1.1)
where f1, ..., fn ∈ h and ℘n denotes the permutation group of order n. Define
⊗ns h = Sn(⊗nh) with ⊗0sh = C. The space
F (h) =∞⊕n=0
⊗ns h, (1.2)
is called the boson Fock space over h. We simply denote F (h) by F . The boson
Fock space F can be identified with the space of `2-sequences (Ψ(n))n≥0 such that
Ψ(n) ∈ ⊗ns h and ‖Ψ‖2F =
∑∞n=0 ‖Ψ(n)‖2
F <∞. The boson Fock space F is a Hilbert
space endowed with the scalar product
(Ψ,Φ)F =∞∑n=0
(Ψ(n),Φ(n))⊗ns h. (1.3)
The element Ω = (1, 0, 0, ...) ∈ F is called Fock vacuum. In the description of
the free quantum field the following operators acting in F are used. There are
two fundamental operators, the creation operator denoted by a∗(f), f ∈ h, and the
annihilation operator by a(f), both acting on F , defined by
(a∗(f)Ψ)(n) =
√nSn(f ⊗Ψ(n−1)), n ≥ 1,
0, n = 0(1.4)
5
with domain D(a∗(f)) =
(Ψ(n))n≥0 ∈ F∣∣∑∞
n=1 n‖Sn(f ⊗Ψ(n−1))‖2F <∞
, and
a(f) = (a∗(f))∗. As the terminology suggests, the action of a∗(f) increases the
number of bosons by one, while a(f) decreases it by one. Since one is the adjoint
operator of the other, the relation (Φ, a(f)Ψ)F = (a∗(f)Φ,Ψ)F holds. Furthermore,
since both operators are closable by the dense definition of their adjoints, we will
use and denote their closed extensions by the same symbols. Let D ⊂ h be a dense
subset. It is known that
F = L.H.a∗(f1) · · · a∗(fn)Ω,Ω| fj ∈ D, j = 1, .., n, n ≥ 1, (1.5)
where L.H. is a shorthand for the linear hull, and · · · denotes the closure in F .
The space
Ffin =
(Ψ(n))n≥0 ∈ F∣∣ Ψ(m) = 0 for all m ≥M with some M
(1.6)
is called finite particle subspace. The field operators a, a∗ leave Ffin invariant and
satisfy the canonical commutation relations
[a(f), a∗(g)] = (f , g)h1l, [a(f), a(g)] = 0, [a∗(f), a∗(g)] = 0 (1.7)
on Ffin. Given a bounded operator T on h, the second quantization of T is the
operator Γ(T ) on F defined by
Γ(T ) =∞⊕n=0
⊗nT. (1.8)
Here it is understood that ⊗0T = 1l. In most cases Γ(T ) is an unbounded operator
resulting from the fact that it is given by a countable direct sum. However, for a
contraction operator T , the second quantization Γ(T ) is also a contraction on F .
The map Γ satisfies
Γ(S)Γ(T ) = Γ(ST ), Γ(S)∗ = Γ(S∗), Γ(1lh) = 1lF . (1.9)
For a self-adjoint operator h on h the structure relations (1.9) imply in particular
that Γ(eith)t∈R is a strongly continuous one-parameter unitary group on F . Then
by the Stone theorem there exists a unique self-adjoint operator dΓ(h) on F such
that
Γ(eith) = eitdΓh, t ∈ R. (1.10)
6
The operator dΓ(h) is called the differential second quantization of h or simply
second quantization of h. Since dΓ(h) = −i ddt
Γ(eith)dt=0, we have
dΓ(h) = 0⊕
∞⊕n=1
n∑j=1
1l⊗ · · ·⊗j
h ⊗ · · · ⊗ 1l︸ ︷︷ ︸n
, (1.11)
where the overline denotes closure, and j on top of h indicates its position in the
product. Thus the action of dΓ(h) is given by
dΓ(h)Ω = 0, (1.12)
dΓ(h)a∗(f1) · · · a∗(fn)Ω =n∑j=1
a∗(f1) · · · a∗(hfj) · · · a∗(fn)Ω. (1.13)
It can be also seen by (1.11) that
σ(dΓ(h)) =
n∑j=1
aj
∣∣∣∣∣ aj ∈ σ(h), j = 1, ..., n, n ≥ 1
∪ 0,
σp(dΓ(h)) =
n∑j=1
aj
∣∣∣∣∣ aj ∈ σp(h), j = 1, ..., n, n ≥ 1
∪ 0.
If 0 6∈ σp(h), the multiplicity of 0 in σp(dΓ(h)) is one. A crucial operator in quantum
field theory is the boson number operator defined by the second quantization of the
identity operator on h: N = dΓ(1lh). Since NΩ = 0 and Na∗(f1) · · · a∗(fn)Ω =
na∗(f1) · · · a∗(fn)Ω, it follows that σ(N) = N ∪ 0. We will use the following facts
below.
Proposition 1.1 Let h be a nonnegative self-adjoint operator, and f ∈ D(h−1/2),
Ψ ∈ D(dΓ(h)1/2). Then Ψ ∈ D(a](f)) and
‖a(f)Ψ‖ ≤ ‖h−1/2f‖‖dΓ(h)1/2Ψ‖, (1.14)
‖a∗(f)Ψ‖ ≤ ‖h−1/2f‖‖dΓ(h)1/2Ψ‖+ ‖f‖‖Ψ‖. (1.15)
In particular, D(dΓ(h)1/2) ⊂ D(a](f)), whenever f ∈ D(h−1/2).
7
To obtain the commutation relations between a](f) and dΓ(h), suppose that f ∈D(h−1/2) ∩D(h). Then
[dΓ(h), a∗(f)]Ψ = a∗(hf)Ψ, [dΓ(h), a(f)]Ψ = −a(hf)Ψ, (1.16)
for Ψ ∈ D(dΓ(h)3/2) ∩ Ffin. By a limiting argument (1.16) can be extended to
Ψ ∈ D(dΓ(h)3/2), and it is seen that a](f) maps D(dΓ(h)3/2) into D(dΓ(h)). In
general we can see that a](f) : D(dΓ(h)n+1/2) → D(dΓ(h)n) for all n ≥ 1, when
f ∈⋂∞n=1D(hn/2). In particular, a](f) maps
⋂∞n=1D(dΓ(h)n) into itself.
Take now h = L2(Rd) and consider the boson Fock space F (L2(Rd)). In this
case, for n ∈ N the space ⊗nsL2(Rd) can be identified with the set of symmetric
functions on L2(Rdn) through
⊗ns L2(Rd) ∼= f ∈ L2(Rdn)|f(k1, . . . , kn) = f(kπ(1), . . . , kπ(n)), ∀π ∈ ℘n. (1.17)
The creation and annihilation operators are realized as
(a(f)Ψ)(n)(k1, ..., kn) =√n+ 1
∫Rdf(k)Ψ(n+1)(k, k1, ..., kn)dk, n ≥ 0, (1.18)
(a∗(f)Ψ)(n)(k1, ..., kn) =
1√n
∑nj=1 f(kj)Ψ
(n−1)(k1, ..., kj, ..., kn), n ≥ 1,
0, n = 0.(1.19)
Here Ψ ∈ F is denoted as a pointwise defined function for convenience, however,
all of these expressions are to be understood in L2-sense. Let ω : L2(Rd)→ L2(Rd)
be the multiplication operator called dispersion relation given by
ω(k) =√|k|2 + ν2, k ∈ Rd, (1.20)
with ν ≥ 0. Here ν describes the boson mass. The second quantization of the
dispersion relation is
(dΓ(ω)Ψ)(n) (k1, ..., kn) =
(n∑j=1
ω(kj)
)Ψ(n)(k1, ..., kn). (1.21)
The self-adjoint operator dΓ(ω) is called the free field Hamiltonian on F (L2(Rd))
and we use the notation
Hf = dΓ(ω). (1.22)
8
The spectrum of the free field Hamiltonian is σ(Hf) = [0,∞), with component
σp(Hf) = 0, which is of single multiplicity with HfΩ = 0. Then formally we may
write the free field Hamiltonian as
Hf =
∫ω(k)a∗(k)a(k)dk. (1.23)
Physically, this describes the total energy of the free field since a∗(k)a(k) gives the
number of bosons carrying momentum k, multiplied with the energy ω(k) of a single
boson, and integrated over all momenta. The commutation relations are
[Hf , a(f)] = −a(ωf), [Hf , a∗(f)] = a∗(ωf). (1.24)
The relative bound of a](f) with respect to the free field Hamiltonian Hf can be
seen from (1.25) and (1.26). If f/√ω ∈ L2(Rd), then
‖a(f)Ψ‖ ≤ ‖f/√ω‖‖Hf
1/2Ψ‖, (1.25)
‖a∗(f)Ψ‖ ≤ ‖f/√ω‖‖Hf
1/2Ψ‖+ ‖f‖‖Ψ‖ (1.26)
hold.
1.2 Segal fields
The creation and annihilation operators are not symmetric and do not commute.
Roughly speaking, a creation operator corresponds to 1√2(x − d
dx) and an annihila-
tion operator to 1√2(x + d
dx) in L2(R). We can, however, construct symmetric and
commutative operators by combining the two field operators and this leads to Segal
fields. The Segal field Φ(f) on the boson Fock space F (h) is defined by
Φ(f) =1√2
(a∗(f) + a(f)), f ∈ h, (1.27)
and its conjugate momentum by
Π(f) =i√2
(a∗(f)− a(f)), f ∈ h. (1.28)
Here f denotes the complex conjugate of f . By the above definition both Φ(f) and
Π(g) are symmetric, however, not linear in f and g over C. Note that, in contrast,
9
they are linear operators over R. It is straightforward to check that [Φ(f),Π(g)] =
iRe(f, g)h1lh, [Φ(f),Φ(g)] = iIm(f, g)h1lh and [Π(f),Π(g)] = iIm(f, g)h1lh. In partic-
ular, for real-valued f and g the canonical commutation relations become
[Φ(f),Π(g)] = i(f, g)h1lh, [Φ(f),Φ(g)] = [Π(f),Π(g)] = 0. (1.29)
Applying the inequalities (1.25) and (1.26) to h = 1l, we see that Ffin is the set of
analytic vectors of Φ(f), i.e., limm→∞
m∑n=0
‖Φ(f)nΨ‖tn/n! <∞ for Ψ ∈ Ffin and t ≥ 0.
The following is a general result.
Proposition 1.2 (Nelson’s analytic vector theorem) Let K be a symmetric
operator on a Hilbert space. Assume that there exists a dense subspace D ⊂ D(K)
such that limm→∞
m∑n=0
‖Knf‖tn/n! < ∞, for f ∈ D and some t > 0. Then K is
essentially self-adjoint on D, and e−tKΦ = s-limm→∞
m∑n=0
tnKnf/n! follows for f ∈ D.
By Nelson’s analytic vector theorem both Φ(f) and Π(g) are essentially self-adjoint
on Ffin. We keep denoting the closures of Φ(f)dFfinand Π(g)dFfin
by the same
symbols.
1.3 Wick product
Loosely speaking, the so-called Wick product :a](f1) · · · a](fn): is defined in a product
of creation and annihilation operators by moving the creation operators to the left
and the annihilation operators to the right without taking the commutation relations
into account. The Wick product :∏n
i=1 Φ(gi): is recursively defined by the equalities
:Φ(f):= Φ(f), :Φ(f)n∏i=1
Φ(fi):= Φ(f) :n∏i=1
Φ(fi): −1
2
n∑j=1
(f, fj) :∏i 6=j
Φ(fi): .
By the above definition we have
:Φ(f)n:=
[n/2]∑k=0
n!
k!(n− 2k)!Φ(f)n−2k
(−1
4‖f‖2
)k. (1.30)
10
Note that :Φ(f1) · · ·Φ(fn): Ω = 2−n/2a∗(f1) · · · a∗(fn)Ω. From this(:n∏i=1
Φ(gi): Ω, :m∏i=1
Φ(fi): Ω
)= δnm2−n/2
∑π∈Pn
n∏i=1
(gi, fπ(i)) (1.31)
follows. The Wick product of the exponential can be computed directly to yield
:eαΦ(f): Ω = e−(1/4)α2‖f‖2eαΦ(f)Ω. (1.32)
Hence for real-valued f and g,
(Ω, eαΦ(f)Ωb) = e(1/4)α2‖f‖2 , α ∈ C. (1.33)
For example (Ω, eiΦ(f)Ωb) = e−(1/4)‖f‖2 and (Ω, eΦ(f)Ωb) = e(1/4)‖f‖2 .
11
12
2 Symplectic structure
2.1 Infinite dimensional symplectic group
In this section we investigate an infinite dimensional symplectic group and its projec-
tive unitary representation on a Fock space. Symplectic transformations leave canon-
ical commutation relations invariant. By symplectic transform of the annihilation
operators and the creation operators a] we can construct operators b] satisfying
the same canonical commutation relations. However it is not necessarily unitarily
equivalent with each others if the dimension of the configuration space is infinity. We
will see it in Proposition 2.1. We will also give an application of symplectic group in
Section 4 to study the spectrum of some quadratic self-adjoint operator. The general
reference of this section is [Ara91, Ber66, HI04, Rui77, Rui78, Seg70, Sha62].
Let C be a conjugation on h, i.e., C is an antilinear isometry on h with C2 = 1l.
For f ∈ h and T ∈ B(h) (B = B(h) is the set of bounded linear operators on h), we
define f ∈ h and T ∈ B(h) by f = Cf and T = CTC . Let I2 = I2(h) denote the
set of Hilbert-Schmidt operators on h. We denote the norm (resp. Hilbert-Schmidt
norm) of a bounded operator X on h by ‖X‖ (resp. ‖X‖2). For S, T ∈ B we define
A =
(S TT S
): h⊕ h→ h⊕ h (2.1)
by
A(φ⊕ ψ) = (Sφ+ Tψ)⊕ (Tφ+ Sψ). (2.2)
Let
J =(
1l 00 −1l
). (2.3)
Then J(φ⊕ ψ) = φ⊕ (−ψ). We define a symplectic group Sp by
Sp =
A =
(S TT S
)∣∣∣∣AJA∗ = A∗JA = J
(2.4)
and the subgroup Sp2 ⊂ Sp by
Sp2 =
A =
(S TT S
)∈ Sp
∣∣∣∣T ∈ I2
. (2.5)
Here A∗ is the adjoint of A, i.e.,
A∗ =(S∗ T ∗
T ∗ S∗
).
13
Note also that the inverse of A ∈ Sp is given by
A−1 = JA∗J =(S∗ −T ∗−T ∗ S∗
).
We can see that A induces the following maps:(a(f)a∗(f)
)7→(bA(f)b∗A(f)
)=
(a∗(Tf) + a(Sf)a∗(Sf) + a(T f)
). (2.6)
Crucial fact is that the map leaves both canonical commutation relations
[bA(f), b∗A(g)] = (f, g)1l, [bA(f), bA(g)] = 0 = [b∗A(f), b∗A(g)], (2.7)
and adjoint relation (Ψ, b∗A(f)Φ) = (bA(f)Ψ,Φ). Furthermore a] can be represented
in terms of b]A:
a(f) = bA(S∗f)− b∗A(T ∗f), (2.8)
a∗(f) = −bA(T ∗f) + b∗A(S∗f). (2.9)
In particular the Segal field and its conjugate are represented as
φ(f) =1√2
(b∗A(S∗f − T ∗f) + bA(S∗f − T ∗f)), (2.10)
π(f) =i√2
(b∗A(S∗f + T ∗f)− bA(S∗f + T ∗f)). (2.11)
We will see below that b]A(f) and a](f) are unitarily equivalent if and only if T
is Hilbert Schmidt operator and will construct the unitary operator implementing
this unitary equivalence.
Proposition 2.1 (Necessary and sufficient condition to the unitary equiv-
alence) Let A =
(S TT S
)∈ Sp2 and define b]A(f) by (2.6). Then there exists
a unitary operator U such that U−1b]A(f) = a](f) if and only if T is the Hilbert
Schmidt operator.
Proof: We give the proof of the necessary part only for the case of h = L2(Rd). The
proof of sufficient part is given in Proposition 2.5.
14
Set ΩA = UΩ. Then bA(f)ΩA = 0 for all f ∈ L2(Rd). Hence (a(Sf) +
a∗(Tf))ΩA = 0 for all f ∈ L2(Rd). Let Pn be the projection from F to the n-
particle subspace. Then we have
a(Sf)Pn+2ΩA = Pn+1a(Sf)ΩA = −Pn+1a∗(Tf)ΩA = −a∗(Tf)PnΩA.
When PmΩA = 0, a(Sf)Pm+2ΩA = 0 for all f ∈ L2(Rd), and then a(f)Pm+2ΩA = 0,
since S−1 exists. Hence PmΩA = 0 implies that Pm+2ΩA = 0. Since bA(f)ΩA = 0,
a(Sf)P1ΩA = 0 and then PnΩA = 0 for all odd number n. If furthermore P0ΩA = 0,
PmΩA = 0 for all even number m, and it implies that ΩA = 0. Since ΩA 6= 0,
κ = P0ΩA 66= 0 follows. Notice that Φ = P2ΩA is a function belonging to L2(Rd×Rd)
and
a(Sf)P2ΩA = −a∗(Tf)P0ΩA. (2.12)
We see that a(Sf)P2ΩA =√
2∫
(Sf)(k′)Φ(k′, k)dk′ and−a∗(Tf)P0ΩA(k) = κTf(k).
Let KΦ be the Hilbert Schmidt operator defined by KΦf(k) =∫f(k′)Φ(k′, k)dk. We
then conclude that
(Tf)(k) =
√2
κ
∫(Sf)(k′)Φ(k′, k)dk′ = KΦSf(k).
Since S is bounded and KΦ is Hilbert-Schmidt, T is Hilbert-Schmidt operator. 2
2.2 Quadratic operators
Let K ∈ I2 and S ∈ B. Then there exist two orthonormal systems ψn and φnin h and a positive sequence λ1 ≥ λ2 ≥ · · · ≥ 0 such that Kf =
∑∞n=0 λn(ψn, f)φn
with∑∞
n=0 λ2n = ‖K‖2
2, where ‖ · ‖2 denotes the Hilbert-Schmidt norm.
Lemma 2.2 [Rui77, Rui78, Ara90] Let en be an arbitrary complete orthonormal
system of h. Then for Ψ ∈ Ffin, sequencesM∑n=1
λna∗(ψn)a∗(φn)Ψ
,
M∑n=1
λna(ψn)a(φn)Ψ
,
M∑n=1
a∗(en)a(S∗en)Ψ
(2.13)
strongly converge as M →∞.
15
Proof: We check only the convergence of∑M
n=1 λna∗(ψn)a∗(φn)Ψ. The others are
similar or rather simpler. We have
∥∥∥∥∥M∑n=1
λna∗(ψn)a∗(φn)Ψ
∥∥∥∥∥2
=M∑m,n
λnλm(Ψ, a(φn)a(ψn)a∗(ψm)a∗(φm)Ψ)
and
a(φn)a(ψn)a∗(ψm)a∗(φm) = δnma(φn)a∗(φm) + δnma∗(ψm)a(ψn) (2.14)
+ (ψn, φm)a(φn)a∗(ψm) + (φn, ψm)a∗(φm)a(ψn) (2.15)
+ a∗(ψm)a∗(φm)a(ψn)a(φn). (2.16)
We will estimate∑M
m,n λmλn(Φ, (∗)Φ) for (∗) = (2.14), (2.15), (2.16) separately. For
(2.14) we have
M∑m,n
λmλn(Φ, (2.14)Φ)
= λ2n‖a∗(φn)Φ‖2 +
∑n
λ2n‖a(ψn)Φ‖2 ≤ 2
∑n
λ2n‖(N + 1)Φ‖2 = 2‖K‖2
2‖(N + 1l)Φ‖2.
For the first term of∑M
m,n λmλn(Φ, (2.15)Φ) we have
M∑m,n
λmλn(ψn, φm)(φn, ψm)‖Φ‖2 +M∑m,n
λmλn(ψn, φm)(a(ψm)Φ, a(φn)Φ).
We see that
M∑m,n
λmλn(ψn, φm)(φn, ψm)‖Φ‖2 =M∑m,n
λmλn((φm, ψn)φn, ψm)‖Φ‖2
=∑m
λm(Kφm, ψm)‖Φ‖2 ≤ (∑n
λ2n)1/2(
∑m
‖Kφm‖2)1/2‖Φ‖2 = ‖K‖22‖‖Φ‖2.
16
For the second term let Φ = a∗(f1) · · · a∗(fL)Ω. We have
M∑m,n
λmλn(ψn, φm)a∗(ψm)a(φn)a∗(f1) · · · a∗(fL)Ω
=∑j
M∑m,n
λmλn(ψn, φm)a∗((φn, fj)ψm)a∗(f1) · · · a∗(fj) · · · a∗(fL)Ω
=∑j
M∑m,n
λmλna∗((φn, fj)(ψn, φm)ψm)a∗(f1) · · · a∗(fj) · · · a∗(fL)Ω
=∑j
∑m
λma∗(λm(Kφm, fj)ψm)a∗(f1) · · · a∗(fj) · · · a∗(fL)Ω
=∑j
a∗(K∗K∗fj)a∗(f1) · · · a∗(fj) · · · a∗(fL)Ω.
Then the second term converges. The second term of of∑M
m,n λmλn(Φ, (2.15)Φ) is
similarly estimated. Finally for∑M
m,n λmλn(Φ, (2.16)Φ), we have
M∑m,n
λmλn(Φ, (2.16)Φ) =∑n
λna(ψn)a(φn)a∗(f1) · · · a∗(fL)Ω
=∑j
a(∑n
λn(φn, fj)ψn)a∗(f1) · · · a∗(fj) · · · a∗(fL)Ω
=∑j
a(K∗fj)a∗(f1) · · · a∗(fj) · · · a∗(fL)Ω.
Then∑M
m,n λmλn(Φ, (2.16)Φ) also converges. 2
We can define for Ψ ∈ Ffin
∆∗KΨ = s− limM→∞
M∑n=1
λna∗(ψn)a∗(φn)Ψ, (2.17)
∆KΨ = s− limM→∞
M∑n=1
λna(ψn)a(φn)Ψ, (2.18)
NSΨ = s− limM→∞
M∑n=1
a∗(en)a(S∗en)Ψ. (2.19)
17
Let Ψ = a∗(f1) · · · a∗(fn)Ω. Then it is seen that
∆KΨ =∑i 6=j
(fi, (K +K∗)fj)a∗(f1) · · · a∗(fi) · · · a∗(fj) · · · a∗(fn)Ω, (2.20)
NSΨ =n∑j=1
a∗(f1) · · · a∗(Sfj) · · · a∗(fn)Ω. (2.21)
We note that (∆K)∗ = ∆∗K∗ and (NS)∗ = NS∗ . Set KT = K∗. It can be also checked
that on Ffin,
[∆∗K , a(f)] = −a∗((K +KT )f), (2.22)
[∆K , a∗(f)] = a((K +KT )f), (2.23)
[NS, a(f)] = −a(STf), (2.24)
[NS, a∗(f)] = a∗(Sf). (2.25)
From (2.22) and (2.23) it follows that
‖∆∗KΩ‖2 = (∆∗KΩ,∆∗KΩ) =∑n
λn(Ω,∆∗Ka∗(ψn)a∗(φn)Ω)
=∑n
λn(Ω, a((K +KT )ψn)a∗(φn)Ω) =∑n
λn((K +KT )ψn, φn).
Since tr(KT ) =∑
n λn(ψ, Tφn), we have ‖∆∗KΩ‖2 = tr(K(K +KT )). Moreover∥∥∥∥∥N∑n=0
1
n!
(−1
2∆∗K
)nΩ
∥∥∥∥∥2
=N∑n=0
an,
where an = (2nn!)−2‖(∆∗K)nΩ‖2. We set D∞ =⋂∞k=1D(Nk). We introduce a subset
I2(h) ⊂ I2(h) by
I2(h) = K ∈ I2(h)|K = KT , ‖K‖ < 1. (2.26)
Proposition 2.3 Let K ∈ I2(h). Then (1) and (2) hold:
(1) For all |z| ≤ ‖K‖−2 and k ≥ 0, the limit limN→∞
N∑n=0
nkanzn exists. In particular
∞∑n=0
anzn = det(1l− zK∗K)−
12 .
18
(2) For all Φ ∈ Ffin, the strong limit
exp
(−1
2∆∗K
)Φ = s− lim
N→∞
N∑n=0
1
n!
(−1
2∆∗K
)nΦ, (2.27)
exists and belongs to D∞.
Proof: Let an,N =
∥∥∥∥∥ 1
n!
(−1
2
N∑m=1
λna∗(φm)a(ψm)
)n
Ω
∥∥∥∥∥2
. Then we can see that
∞∑n=0
an,Nαn =
1√∏Nj=1(1− αλ2
j)
for |α| < 1. By the limiting argument we have
∞∑n=0
anαn =
1√∏∞j=1(1− αλ2
j)= [det(1l− αK∗K)]−1/2.
In particular∑∞
n=k n(n− 1) · · · (n− k + 1)anαn−k <∞. Thus∥∥∥∥∥
∞∑n=k
√n(n− 1) · · · (n− k + 1)
1
n!(−1
2∆K)nΩ
∥∥∥∥∥2
<∞
and∞∑n=0
1
n!(−1
2∆K)nΩ ∈ D∞.
Furthermore∞∑n=0
1
n!
(−1
2∆K
)nΦ converges for Φ = a∗(f1) · · · a∗(fL)Ω. 2
Suppose that S ∈ B and K ∈ I2. Then for Ψ ∈ Ffin, we can also see (rather
easier than (2.27)) that
:e−NS:Ψ = s− limM→∞
M∑n=0
1
n!:(−NS)n: Ψ (2.28)
e−12
∆KΨ = s− limM→∞
M∑n=0
1
n!
(−1
2∆K
)nΨ (2.29)
19
exist, and :e−NS:Ψ, e−12
∆KΨ ∈ Ffin. By (2.22)-(2.25) we can check the following
commutation relations on Ffin:
[e−12
∆∗K , a(f)] =1
2a∗((K +KT )f)e−
12
∆∗K , (2.30)
[e−12
∆K , a∗(f)] = −1
2a((K +KT )f)e−
12
∆K , (2.31)
[:e−NS:, a(f)] = a(STf) :e−NS:, (2.32)
[:e−NS:, a∗(f)] = −a∗(Sf) :e−NS: . (2.33)
Corollary 2.4 Let K1 ∈ I2, K2 and K−12 be in B, and K1K
−12 ∈ I2. Then
a∗(K1f) + a(K2f)Ω(K1K−12 ) = 0, f ∈ h, (2.34)
where Ω(K1K−12 ) = exp
(−1
2∆∗K1K
−12
)Ω.
Proof: By (2.30), we can see that
a(K2f) exp
(−1
2∆∗K1K
−12
)Ω = −a∗(K1f) exp
(−1
2∆∗K1K
−12
)Ω.
The desired result follows. 2
2.3 Bogoliubov transformations
In this section we construct a unitary operator implementing the unitary equivalence
between a] and b]A when A ∈ Sp2.
2.3.1 Homogeneous case
Let A =
(S TT S
)∈ Sp. Then A induces the following maps:
(a(f)a∗(f)
)7→(bA(f)b∗A(f)
)=
(a∗(Tf) + a(Sf)a∗(Sf) + a(T f)
). (2.35)
Formally we may write as (bA(f) b∗A(f)) = (a(f) a∗(f))A. Since bA(f)dFfin(resp. b∗A)
is closable, we denote its closed extension by the same symbol bA(f) (resp b∗A(f)).
It is seen in Proposition 2.1 that there exists a unitary operator UA on F such that
20
U −1A b#
A(f)UA = a](f) if A ∈ Sp2. The condition
(S TT S
)∈ Sp is equivalent to
the following algebraic relations:
S∗S − T ∗T = 1l, (2.36)
S∗T − T ∗S = 0, (2.37)
SS∗ − TT ∗ = 1l, (2.38)
TS∗ − ST ∗ = 0. (2.39)
Using these algebraic relations we can prove that S−1 ∈ B, ‖TS−1‖ < 1, (TS−1)T =
TS−1, and (S−1T )T = S−1T . We set
K1 = TS−1, K2 = 1l− (S−1)∗, K3 = −S−1T . (2.40)
Let
(S TT S
)∈ Sp2. Since K1 ∈ I2, KT
1 = K1 and ‖K1‖ < 1, i.e., K1 ∈ I2, we can
see that by Proposition 2.3,
UA = det(1l−K∗1K1)1/4e−12
∆∗K1 :e−NK2:e−12
∆K3 (2.41)
is well defined on Ffin and UAΨ ∈ D∞ for Ψ ∈ Ffin. UA is called the intertwining
operatorassociated with A ∈ Sp2.
Proposition 2.5 (Homogeneous case) Let A ∈ Sp2. Then UA can be uniquely
extended to the unitary operator on F and
U −1A b]A(f)UA = a](f) (2.42)
holds for all f ∈ h.
Proof: Let U1 = e−12
∆∗K1 , U2 = :e−NK2: and U3 = e−12
∆K3 . By commutation relations
(2.30)-(2.33) we see that
U1U2U3a∗(f) = U1U2a
∗(f)U3 + U1U2a(−K3f)U3
= a∗((1l−K2)f)U1U2U3 + U1U2a(−K3f)U3. (2.43)
Using 1l − K2 = (S−1)∗ = S − TS−1T and a∗((1l − K2)f) = b∗A(f) + a(−Tf) +
a∗(−TS−1Tf), we have
a∗((1l−K2)f)U1U2U3
= b∗A(f)U1U2U3 + U1a∗(−TS−1Tf)U2U3 + U1a∗(K1Tf) + a(−Tf)U2U3.
21
Hence the right hand side above is identical with
a∗((1l−K2)f)U1U2U3 = b∗A(f)U1U2U3 + U1U2a(−Tf) + a(KT2 Tf)U3. (2.44)
Combining (2.43) and (2.44), we obtain
U1U2U3a∗(f) = b∗A(f)U1U2U3 + U1U2a(−Tf +KT
2 Tf −K3f)U3.
Since −T +KT2 T−K3 = 0, we get U1U2U3a
∗(f)Φ = b∗A(f)U1U2U3Φ for all Φ ∈ D∞and f ∈ h. I.e.,
UAa](f)Φ = b#
A(f)UAΦ, Φ ∈ D∞. (2.45)
From this, and the canonical commutation relations it follows that
‖UAa∗(f1) · · · a∗(fn)Ω‖2 = ‖b∗A(f1) · · · b∗A(fn)UAΩ‖2 = ‖a∗(f1) · · · a∗(fn)Ω‖2,
where we used that ‖e−12
∆∗K1 Ω‖2 = det(1l−K∗1K1)−1/2 and bA(f)e−12
∆∗K1 Ω = 0. Then
UA is an isometry from Ffin onto the dense subspace:
E = L.H.b∗A(f1) · · · b∗A(fn)UAΩ,UAΩ|fj ∈ h, j = 1, ..., n, n ≥ 1.
We notice that b#A(f)E ⊂ E and a] can be represented in terms of a linear combi-
nation of b#A . See (2.8) and (2.9). By this we see that a](f) also leaves E invariant:
a](f)E ⊂ E for all f ∈ h. Let Ψ ∈ E and ΨN = Ψ(0),Ψ(1), ...,Ψ(N), 0, 0, .... Since
ΨN ∈ Ffin, we see that ΨN is an analytic vector of φ(f) = 1√2(a∗(f) + a(f)), which
implies, together with a](f)E ⊂ E , that eiφ(f)ΨN ∈ E , and by a limiting argument
eiφ(f)Ψ ∈ E . Thus eiφ(f)E ⊂ E follows. By a limiting argument we have eiφ(f)E ⊂ E .
Thus E = F by the irreducibility of φ(f). Hence we conclude that UA can be
uniquely extended to a unitary operator on F . Then the proposition follows. 2
2.3.2 Inhomogeneous case
Let A =
(S TT S
)∈ Sp and L ∈ h. Then it induces the map
(a(f)a∗(f)
)7→(bA,L(f)b∗A,L(f)
)=
(a∗(Tf) + a(Sf) + (L, f)a∗(Sf) + a(T f) + (L, f)
). (2.46)
22
It is clear that b#A,L satisfies canonical commutation relations: [bA,L(f), b∗A,L(g)] =
(f , g) and [bA,L(f), bA,L(g)] = 0 = [b∗A,L(f), b∗A,L(g)]. Moreover a] can be represented
in terms of a linear sum of b#A,L:
a(f) = bA,L(S∗f)− b∗A,L(T ∗f) + (−SL+ T L, f), (2.47)
a∗(f) = −bA,L(T ∗f) + b∗A,L(S∗f) + (−SL+ TL, f). (2.48)
We define the operator π(L) in F by π(L) = ibA(L) − b∗A(L), which can be
represented in terms of a] by
π(L) = ia∗(TL− SL)− a(TL− SL). (2.49)
Let SA,L = exp(−iπ(L)) = exp(bA(L)− b∗A(L)) and we define UA,L by
UA,L = SA,LUA. (2.50)
Here SA,L is called the displacement operator and UA.L Bogoliubov transform.
Proposition 2.6 (Inhomogeneous case) Let A ∈ Sp2 and L ∈ h. Then we have
U −1A,Lb
]A,L(f)UA,L = a](f). (2.51)
Proof: Notice that SA,Lb#A(f)S−1
A,L = b]A,L(f). Then the proposition follows from
Proposition 2.5. 2
Suppose that A ∈ Sp2 and L ∈ h. Let Φ = UA,LΩ. For later use we compute
(a∗(f)Ω,Φ) and (a∗(f)a∗(g)Ω,Φ).
Lemma 2.7 Suppose that A =
(S TT S
)∈ Sp2 and L ∈ h. Set ξ = TL− SL and
K = TS−1. Then it follows that
(a∗(f)Ω,Φ)
(Ω,Φ)= (f, ξ) + (Kf, ξ), (2.52)
(a∗(f)a∗(g)Ω,Φ)
(Ω,Φ)
= (f, ξ)(g, ξ) + (g, ξ)(Kf, ξ) + (f, ξ)(Kg, ξ) + (Kg, ξ)(Kf, ξ)− (f,Kg). (2.53)
23
Proof: We set S = SA,L = exp(a∗(ξ) − a(ξ)),U = UA and J = (Ω,Φ). Notice that
S−1a(f)S = a(f) + (f , ξ) and Sa(f)S−1 = a(f)− (f , ξ). Then directly we have
(a∗(f)Ω,Φ) = (Ω, a(f)SU Ω)
= (f, ξ)J + (Ω, Sa(f)U Ω)
= (f, ξ)J + (S−1Ω,−a∗(Kf)U Ω)
= (f, ξ)J + (−a(Kf)S−1Ω,U Ω)
= (f, ξ)J + (Kf, ξ)J.
Then (2.52) follows. Next we see that
(a∗(f)a∗(g)Ω,Φ)
= (Ω, a(g)a(f)Φ)
= (S−1Ω, (a(g) + (g, ξ))(a(f) + (f, ξ))U Ω)
= (S−1Ω, a(g)a(f)U Ω) + (g, ξ)(S−1Ω, a(f)U Ω)
+ (f, ξ)(S−1Ω, a(g)U Ω) + (g, ξ)(f, ξ)J
= (g, ξ)(f, ξ)J + (g, ξ)(Kf, ξ)J + (f, ξ)(Kg, ξ)J + (S−1Ω, a(g)a(f)U Ω)J.
Moreover we have
(S−1Ω, a(g)a(f)U Ω)
= (S−1Ω, a(g)(−a∗(Kf))U Ω)
= (S−1Ω,−a∗(Kf)a(g)U Ω)− (f,Kg)J
= (S−1Ω, a∗(Kf)a∗(Kg)U Ω)− (f,Kg)J
= (a(Kf)a(Kg)S−1Ω,U Ω)− (f,Kg)J
= (S−1(a(Kf)− (Kf, ξ))(a(Kg)− (Kg, ξ))Ω,U Ω)− (f,Kg)J
= (Kf, ξ)(Kg, ξ)J − (f,Kg)J.
Then (2.53) follows. 2
Lemma 2.8 Suppose that A =
(S TT S
)∈ Sp2 and L ∈ h. Set ξ = TL− SL and
K = TS−1. We assume that ξ = ξ and f = f . Then
((2a∗(f) + a∗(f)a∗(f))Ω,Φ) = (2γ + γ2 − (f,Kf))(Ω,Φ), (2.54)
24
where γ = (ξ, (1l +K)f). In particular for all p ∈ R, it follows that
((p+ a(f) + a∗(f))2Ω,Φ)
(Ω,Φ)= (p+ γ)2 + (f, (1l−K)f). (2.55)
Proof: By the assumptions we have
(a∗(f)Ω,Φ) = γ,
(a∗(f)a∗(f)Ω,Φ) = (f, ξ)2 + 2(ξ, f)(ξ,Kf) + (ξ,Kf)2 − (f,Kf) = γ2 − (f,Kf).
Then (2.54) follows. Notice that
(p+ a(f) + a∗(f))2Ω = (p2 + 2pa∗(f) + a∗(f)a∗(f))Ω + ‖f‖2Ω.
Then (2.55) follows from (2.54). 2
2.4 One parameter symplectic groups and 2-cocycles
In this section we review the pseudo unitary representation of symplectic groups
Sp2. Let U (F ) be the set of unitary operators on F . Then we can define the map
U· : Sp2 → U(F ). (2.56)
Since
U −1AB UAUBa
](f) = a](f)U −1AB UAUB
for all f ∈ h, it follows that U −1AB UAUB = ω(A,B)1l with ω(A,B) ∈ U(1). Thus U·
gives a projective unitary representation of Sp2, i.e.,
UAUB = ω(A,B)UAB (2.57)
where ω(A,B) ∈ U(1) is 2-cocycle. Let
sp =
A =
(S TT S
)∣∣∣∣AJ + JA∗ = A∗J + JA = 0
(2.58)
=
A =
(S TT S
)∣∣∣∣S∗ = −S, T ∗ = T
. (2.59)
25
If A ∈ sp, then it can be shown that
etA ∈ Sp, t ∈ R. (2.60)
(2.60) is called the one-parameter symplectic group. Set
sp2 =
A =
(S TT S
)∈ sp
∣∣∣∣T ∈ I2
. (2.61)
If A ∈ sp2, then we can also show that
etA ∈ Sp2, t ∈ R. (2.62)
Let Ut = UetA be the intertwining operator associated with A ∈ sp2. Thus Ut
satisfies that
UtUs = eiρ(t,s)Ut+s (2.63)
with the so called local exponent ρ(t, s) ∈ R. The local exponent ρ(t, s) also satisfies
relation:
ρ(t, s) + ρ(t+ s, r) = ρ(s, r) + ρ(t, s+ r). (2.64)
We shall show the explicit form of the local exponent. Let A =
(S TT S
)∈ sp2.
Then εtA =
(St T tTt St
)∈ Sp2 induces the map
(a(f)a∗(f)
)7→(bt(f)b∗t (f)
)=
(a∗(Ttf) + a(Stf)a∗(Stf) + a(Ttf)
). (2.65)
The intertwining operator Ut = UetA implements the map (2.65), i.e.,
U −1t b]t(f)Ut = a](f) (2.66)
for all f ∈ h. Furthermore we see that
d
dtbt(f) = a(Sf) + a∗(Tf) (2.67)
d
dtb∗t (f) = a(Sf) + a∗(T f). (2.68)
26
Let A =
(S TT S
)∈ sp2. We define
∆(A) =i
2(∆∗T −∆T )− iNS. (2.69)
Operator ∆(A) is essentially self-adjoint on Ffin and we can also directly see com-
mutation relations:
[i∆(A), a(f)] = a(Sf) + a∗(Tf), (2.70)
[i∆(A), a∗(f)] = a(T f) + a∗(Sf). (2.71)
By using commutation relations we have
eit∆(A)a](f)e−it∆(A) = b]t(f) (2.72)
and hence the equation Uteit∆(A)a](f) = a](f)Ute
it∆(A) is derived, i.e.,
[Uteit∆(A), a](f)] = 0.
Thus for A ∈ sp2 there exists θA(t) ∈ R such that
Ut = eiθA(t)e−it∆(A). (2.73)
It is immediate from (2.73) that [Ut,∆(A)] = 0.
Lemma 2.9 Let A ∈ sp2. Then the function θA(·) is C1(R).
Proof: By the definition of θ(t) we have
eiθA(t) =(Ω,UtΩ)
(Ω, eit∆(A)Ω).
We can check that (Ω,UtΩ) and (Ω, eit∆(A)Ω) are differentiable in t. Then the lemma
follows. 2
Proposition 2.10 (Local exponent) Let A =
(S TT S
)∈ sp2 and we set etA =(
St T tTt St
). Then
27
(1) θ(t) =
∫ t
0
τrdr and τr = 12Imtr(T ∗TrS
−1r ),
(2) ρ(t, s) =
∫ t
0
τrdr +
∫ s
0
τrdr −∫ s+t
0
τrdr.
Proof: Let Kt = TtS−1t . We notice that R 3 det(1l − K∗tKt)
1/4 = (Ω,UtΩ) =
(Ω, eiθA(t)e−it∆(A)Ω). Thus we see that
d
dt(Ω, eiθA(t)e−it∆(A)Ω) = iθ′A(t)(Ω,UtΩ) + (Ω, i∆(A)UtΩ) ∈ R
which implies that the imaginary part of the right hand side disappear and then
θ′A(t)(Ω,UtΩ) = −Im(Ω, i∆(A)UtΩ) = −Im(1
2∆∗TΩ,∆∗KtΩ).
From this we have
θ′A(t) = −Im1
2
(∆∗TΩ,UtΩ)
(Ω,UtΩ)= −1
2Im(∆∗TΩ, e−
12
∆∗KtΩ) =1
4Im(∆∗TΩ,∆∗KtΩ).
Since (∆∗TΩ,∆∗KtΩ) = (Ω, [∆T ∗ ,∆∗Kt
]Ω) = 2tr(T ∗Kt), we complete (1). The state-
ment (2) follows from (1) immediately. 2
From this proposition 2-cocycle eiρ(t,s) vanishes if Imtr(T ∗TrS−1r ) = 0. We give
a sufficient condition to vanish the 2-cocycle.
Corollary 2.11 Suppose A =
(S TT S
)∈ sp2 and S = S = −S∗ and T = T = T ∗.
Then ρ(t, s) = 0, i.e., Ut = eit∆(A). In particular Ut, t ∈ R, is the one-parameter
unitary group.
Example 2.12 Let h = L2(R). Define the Hilbert-Schmidt operator T by Tf(x) =∫K(x, y)f(y)dy with a real-valued function K ∈ L2(R × R). Let h be a real-
valued function such that h ∈ L∞(R) and h(−x) = −h(x). Define the operator
S = h(d/dx). Then S and T satisfy the condition in Corollary (2.11).
28
Part II
The Pauli-Fierz model
3 The Pauli-Fierz Hamiltonian
3.1 Introduction
It was well known that vacuum polarization divergences, self-energy divergences and
another infinity plagued QED in 1930’s. When one attempted to compute calculate
the contribution of radiative effects to the scattering of electrons by the Coulomb
field of a nucleus, infrared divergences were encountered. In 1937 Bloch and Nord-
sieck [BN37] showed that this infrared divergence arose from the illegitimate neglect
of processes involving the simultaneous emission of many photons, i.e, an emission
of photons of very low frequencies yields divergences of an electromagnetic mass, a
scattering cross section, etc. In 1938 according to a certain model describing an in-
teraction between an electron and a quantized radiation field Pauli and Fierz[PF38]
recognized that the quantized radiation field reacts back on the electron to produce
an electromagnetic mass. This model is today the so-called Pauli-Fierz model, which
is the main object in this paper1. The concept of mass renormalization in QED has
its origin in these researches of Pauli and Fierz.
Here we look at a typical example of successes of the Pauli-Fierz model. Let
αµ, µ = 1, 2, 3, and β are 4 × 4 Hermitian matrices obeying the anticommutation
relations αµ, αν = 2δµν1, αµ, β = 0 and β2 = 1. The Dirac Hamiltonian of a
hydrogen-like atom is given by
D =3∑
µ=1
αµ(−i∇µ) + βm− Ze2
|x|, (3.1)
where Z is an atomic number, e the charge of an electron, and m the mass of an
electron. Then D has eigenvalues
Enj =m√
1 + Z2e4(n− (j + 1/2) +
√(j + 1/2)2 − Z2e4
)−2, (3.2)
1In this note we take the dipole approximation of the standard Pauli-Fierz Hamiltonian.
29
Figure 1: Electron fluctuated by radiation in hydrogen atom
where n = 1, 2, ..., denotes the principal quantum number and j = l ± 1/2 the total
angular-momentum with the angular-momentum l = 0, ..., n − 1. Eigenvalue 2S1/2
corresponds to n = 2, j = 1/2, l = 0, and 2P1/2 to n = 2, j = 1/2, l = 1. Then the
Dirac theory concludes that two levels 2S1/2 and 2P1/2 in a hydrogen-like atom sit at
the same energy level, i.e.,
2S1/2 = 2P1/2.
In 1947, by Lamb and Retherford[LR47], it was experimentally observed, however,
that
2S1/2 > 2P1/2.
This discrepancy is called the Lamb-shift. Bethe[Bet47] regarded the Lamb-shift as
an evidence of a radiation reaction, and tentatively made a nonrelativistic calculation
of the difference of the two levels. The resulting value was in remarkable agreement
with the observation. In 1948, using the Pauli-Fierz model, Welton[Wel48] gave an
intuitive derivation to the Lamb-shift. He argued that a position-fluctuation of an
electron through the radiation field will effectively modify the external potential V
(Figure 1). The fluctuation was thought of as a Gaussian random variable ∆x, then
an effective potential is formally given by a mean value of V (x+ ∆x);
Veff(x) = 〈V (x+ ∆x)〉AVE = (2πC)−3/2
∫R3
V (y)e−|x−y|2/(2C)dy, (3.3)
with a certain positive constant C. Then an electron Hamiltonian effectively turns
out to be governed by a Hamiltonian with the external potential Veff instead of
30
V . Welton gave an interpretation of the Lamb shift as the difference between the
spectrum of the original Hamiltonian and an effective one.
In Part II we study the Pauli-Fierz Hamiltonian. The basic assumptions are as
follows.
(1) We take the dipole approximation.
(2) We neglect spin.
The spectrum of this model is studied by the series of papers by A. Arai [Ara81-a,
Ara81-b, Ara81-c, Ara83-a, Ara83-b]. Since we take the dipole approximation, the
Pauli-Fierz Hamiltonian H is reduced to simple. Although it is not translation
invariant, i.e., it does not commute with the total momentum, it commutes with
particle momentum. Then H without external potential can be decomposable with
respect to the momentum of the particle H =∫ ⊕Rd Hpdp and we can diagonalize Hp
for each fiber p ∈ Rd by applying Bogoliubov transform studied in Section 2. This
is a key observation in this section.
3.2 The Pauli-Fierz Hamiltonian with the dipole approxi-mation
Let us assume that the electron moves in dimension d ≥ 3. The physically reasonable
dimension is d = 3. We denote by F the boson Fock space over the one particle
space L2(Rd × 1, ..., d − 1). Here a photon is regarded as a transversal wave in
d− 1 directions. The Hilbert space H of the coupled system is then given by
H = L2(Rd)⊗F . (3.4)
The annihilation operator a(f, j) and the creation operator a∗(g, j) satisfies canon-
ical commutation relation:
[a(f, j), a∗(g, j′)] = δjj′(f, g)L2(Rd), [a(f, j), a(g, j′)] = 0 = [a∗(f, j), a∗(g, j′)]
(3.5)
on the finite particle subspace Ffin for f, g ∈ L2(Rd) and 1 ≤ j, j′ ≤ d − 1. The
d-dimensional polarization vectors are written as
ej(k) = (ej1(k), . . . , ejd(k)), j = 1, . . . , d− 1, (3.6)
31
which satisfy ei(k) · ej(k) = δij and ej(k) · k = 0 almost everywhere on Rd. Let
Hf = dΓ(ω) (3.7)
be the free field Hamiltonian with the dispersion relation
ω(k) = |k|. (3.8)
Let2
Aµ(η) =1√2
∫ejµ(k)√ω(k)
(η(−k)a∗(k, j) + η(k)a(k, j)) dk,
Πµ(η) =i√2
∫ √ω(k)ejµ(k) (η(−k)a∗(k, j)− η(k)a(k, j)) dk.
When η is real, then η(k) = η(−k) and Aµ(η) and πµ(η) are symmetric. We have
the following commutation relations on Ffin
[Aµ(η),Πν(ρ)] = i
∫dµν(k)η(−k)ρ(k)dk = i(dµν ˆη, ρ), (3.9)
[Aµ(η), Aν(ρ)] = 0, (3.10)
[Πµ(η),Πν(ρ)] = 0, (3.11)
where
dµν(k) = δµν −kµkν|k|2
(3.12)
denotes the transversal delta function, and
[Hf , Aµ(η)] = −iΠµ(η), (3.13)
[Hf ,Πµ(η)] = iAµ(−∆η). (3.14)
The quantized radiation field Aµ with a cutoff function ϕ is defined by
Aµ =1√2
∫1√ω(k)
ejµ(k) ˜ϕ(k)a∗(k, j) + ϕ(k)a(k, j)
dk, (3.15)
2Throughout Part II in this lecture note the summation over repeated indices is understood.The Greek letters µ, ν, ... and a, b run from 1 to d, and i, j, k from 1 to d− 1.
32
and the quantized electric field, as its canonically conjugate, by
Πµ = i1√2
∫ √ω(k)ejµ(k)
˜ϕ(k)a∗(k, j)− ϕ(k)a(k, j)dk. (3.16)
Here f(k) = f(−k). They satisfy that
[Aµ,Πν ] = i
∫dµν(k)ϕ(−k)ϕ(k)dk, (3.17)
[Aµ, Aν ] = 0, (3.18)
[Πµ,Πν ] = 0. (3.19)
We define the Pauli-Fierz Hamiltonian.
Definition 3.1 (Pauli-Fierz Hamiltonian!dipole approximation) The Pauli-
Fierz Hamiltonian H with the dipole approximation is defined by
H =1
2m(−i∇⊗ 1l− α1l⊗ A)2 + V ⊗ 1l + 1l⊗Hf , (3.20)
where α ∈ R denotes the coupling constant and V : Rd → R is an external potential.
In what follows we omit the tensor notation ⊗ for notational convenience. Thus H
is simply written as1
2m(−i∇− αA)2 + V +Hf .
We first of all state the self-adjointness of H.
Proposition 3.2 (Self-adjointness) [Ara81-a] Suppose that ϕ/ω,√ωϕ ∈ L2(Rd)
and ϕ(−k) = ϕ(k), and that V is relatively bounded with respect to − 12m
∆ with a
relative bound strictly smaller than one3. Then H is self-adjoint on D(−∆)∩D(Hf)
and bounded below for arbitrary α ∈ R.
Proof: Let V = 0. Let L = −∆ +Hf + 1l. It can be seen that
|(HF,LG)− (LF,HG)| ≤ C‖L1/2F‖‖L1/2G‖
3D(V ) ⊂ D(−∆) and ‖V f‖ ≤ a‖ − 12m∆f‖+ b‖f‖ with some a < 1.
33
with some constant C. Then H is essentially self-adjoint on D(−∆)∩D(Hf) by the
Nelson commutator theorem. Furthermore by the inequality ‖HF‖ ≤ C‖LF‖, the
closedness of HdD(−∆)∩D(Hf) follows. For nonzero V , by the diamagnetic inequality,
‖( 1
2m(−i∇− αA)2 +Hf − z)−1F‖ ≤ ‖(− 1
2m∆ +Hf − z)−1F‖,
we can see that V is also relatively bounded with respect to 12m
(−i∇− αA)2 + Hf
with a relative bound strictly smaller than one. Then the proposition follows by the
Kato-Rellich theorem. 2
Let
Aµ(x) =1√2
∫1√ω(k)
ejµ(k)(ϕ(−k)e−ikxa∗(k, j) + ϕ(k)eikxa(k, j)
)dk
for each x ∈ Rd. Under the identification H =∫ ⊕Rd Fdx, we define
Aµ =
∫ ⊕RdAµ(x)dx.
Thus the Pauli-Fierz Hamiltonian without the dipole approximation is defined by
1
2m(−i∇⊗ 1l− αA)2 + V ⊗ 1l + 1l⊗Hf . (3.21)
The Pauli-Fierz Hamiltonian H under consideration in this lecture note is defined
by (3.21) with Aµ replaced by 1l⊗ Aµ(0).
3.3 Translation invariant Hamiltonian
Suppose that V = 0. Let us define the operator K in H by
K =1
2m(−i∇− αA)2 +Hf . (3.22)
Since K commutes with −i∇µ, the Hilbert space H and the operator K are de-
composable with respect to the joint spectrum of −i∇µ, i.e.
H =
∫ ⊕Rd
Fdp,
K =
∫ ⊕RdHpdp
34
where
Hp =1
2m(p− αA)2 +Hf , p ∈ Rd. (3.23)
Proposition 3.3 (Self-adjointness) [Ara83-a] Suppose that ϕ/ω,√ωϕ ∈ L2(Rd)
and ϕ(−k) = ϕ(k). Then Hp is self-adjoint on D(Hf) and bounded below for arbi-
trary α ∈ R.
Proof: Let L = Hf + 1l. It can be seen that
|(HpF,LG)− (LF,HpG)| ≤ C‖L1/2F‖‖L1/2G‖
with some constant C. Then Hp is essentially self-adjoint on D(Hf) by the Nel-
son commutator theorem. Furthermore by the inequality ‖HpF‖ ≤ C‖LF‖, the
closedness of HpdD(Hf) follows. Then the proposition follows. 2
The quadruple
D = a∗, a,Hf ,Ω
satisfies the algebraic relations: [a(f, j), a∗(g, j′)] = δjj′(f , g), [Hf , a(g)] = −a(ωg),
[Hf , a∗(g)] = a∗(ωg) and HfΩ = a(f)Ω = 0. In this section we construct operators
Bp and B∗p and a vector Ωp such that the quadruple
Dp = Bp, B∗p , Hp − Ep,Ωp
satisfies the same algebraic relations as those of D for each p ∈ Rd, where Epdenotes the ground state energy of Hp, i.e., Ep = infσ(Hp). We need in addition
some technical assumptions on ϕ.
Assumption 3.4 We suppose (1),(2), (3) or (1), (2’), (3):
(1)√ωϕ, ϕ/ω ∈ L2(Rd), ϕ(k) = ϕ(−k) and ϕ is rotation invariant, i.e. ϕ(k) =
ϕ(|k|),
(2) ϕ(k) 6= 0 for k 6= 0, and ρ(s) = |ϕ(√s)|2s d−2
2 ∈ Lε([0,∞), ds) for some 1 < ε,
and there exists 0 < C < 1 such that |ρ(s + h) − ρ(s)| ≤ K|h|C for all s and
0 ≤ h ≤ 1,
(2’) ϕ(k) 6= 0 for λ ≤ |k| ≤ Λ, and ϕ(k) = 0 for |k| > Λ and |k| < λ with some
Λ > 0 and λ > 0,
35
(3) ‖ϕω(d−3)/2‖∞ <∞ and ‖ϕω(d−1)/2‖∞ <∞.
Assumption (1) is used for self-adjointness of H, (2) or (2’) for the definition of a
function D+ and Q in Section 3.4, and (3) for an operator Tµν in Lemma 3.7. The
main theorem in Section 3 is as follows.
Theorem 3.5 (Diagonalization of Hp) Suppose Assumption 3.4.
(1) Let p = 0. Then there exists a unitary operator U0 : D(Hf)→ D(Hf) such that
U −10 H0U0 = Hf + g. (3.24)
(2) Suppose, in addition,
∫ϕ2
ω3dk <∞. Then for all p ∈ Rd, there exists a unitary
operator Up : D(Hf)→ D(Hf) such that
U −1p HpUp =
1
2meff
p2 +Hf + g. (3.25)
Here the effective mass meff is given by
meff = m+ α2
(d− 1
d
)‖ϕ/ω‖2, (3.26)
and the additional constant g by
g =d
2π
∫ ∞−∞
α2(d−1d
) ∥∥ tϕt2+ω2
∥∥2
m+ α2(d−1d
) ∥∥∥ ϕ√t2+ω2
∥∥∥2dt. (3.27)
The condition
∫ϕ2
ω3dk < ∞ is called the infrared regular condition, on the other
hand
∫ϕ2
ω3dk = ∞ infrared singular condition. The term α2
(d−1d
)‖ϕ/ω‖2 in meff is
called self-energy. The self energy α2(d−1d
) ∫ |ϕ(k)|2|k|2 dk has no singularity at the origin
k = 0, since d ≥ 3.
We furthermore define the unitary operator U on H =∫ ⊕Rd Fdp by
U =
∫ ⊕Rd
Upeiπ
2Ndp, (3.28)
where N denotes the number operator in F .
36
Theorem 3.6 (Diagonalization of H) Suppose Assumption 3.4 and that V is
relatively bounded with respect to − 12m
∆ with a relative bound strictly smaller than
one. Assume furthermore
∫ϕ2
ω3dk <∞. Then, for each α ∈ R, U maps D(−∆) ∩
D(Hf) onto itself and
U −1HU = Heff +Hf + δV + g, (3.29)
where Heff denotes the effective Hamiltonian given by
Heff = − 1
2meff
∆ + V,
and δV is the perturbation given by
δV = T−1V T − V, (3.30)
with
T = exp (−i(−i∇) ·K) , (3.31)
Kµ =1√2
∫ejµ(k)√ω(k)
(αϕ(k)
meff(k)ω(k)a∗(k, j) +
αϕ(k)
meff(k)ω(k)a(k, j)
)dk. (3.32)
Here the function meff(k) is given by (3.39) below.
We shall give proofs of Theorems 3.6 and 3.5 in Section 3.5.
Formally
T−1V T (x) = e∇·KV e−∇·K(x) = V (x+K).
Thus
V (x+K) =∞∑n=0
1
n!(∇ ·K)nV (x).
Thus the discrepancy between V and Veff is given by∑∞
n=11n!
(∇ · K)nV (x). In
particular
(Ω,∞∑n=1
1
n!(∇ ·K)nV (x)Ω) = (Ω,
∞∑n=2
1
n!(∇ ·K)nV (x)Ω)
and (Ω,∑∞
n=21n!
(∇·K)nV (x)Ω) ∼ 12(Ω, (∇·K)2V (x)Ω). Approximately the radiative
effect changes V to V + 12(Ω, (∇ · K)2V (x)Ω). This gives an interpretation of the
Lamb shift. See [Bet47, Wel48].
37
3.4 Bogoliubov transformation
3.4.1 Algebraic relations
In order to prove Theorems 3.6 and 3.5 we prepare several lemmas.
Let us consider the time evolution of Aµ(f) by the Hamiltonian Hp. Let
Aµ(f, t) = eitHpAµ(f)e−itHp
for p ∈ Rd, and set Aµ(f, t) =∫Aµ(x, t)f(x)dx. Formally we have
(∂2
∂t2−∆)Aµ(x, t) =
α
m(pν − αAν(x, t))ρνµ(x), (3.33)
where ρνµ(x) = (2π)−d/2∫dνµ(k)ϕ(k)eikxdk. We shall operator-theoretically solve
(3.33) in what follows. Let us define
D(z) = m− α2
(d− 1
d
)∫Rd
|ϕ(k)|2
z − ω(k)2dk, z ∈ C \ [0,∞). (3.34)
Lemma 3.7 Suppose Assumption 3.4.
(1) The function D(z) is analytic and has no zero points in C \ [0,∞).
(2) The function D±(s) = limε↓0
D(s ± iε) exists for all s ∈ [0,∞), D±(0) = meff
and lims→∞
D±(s) = m.
(3) It follows that D+(s)−D−(s) = πiα2
(d− 1
d
)|Sd−1||ϕ(
√s)|2s
d−22 ,
where |Sd−1| = 2πd+1
2 /Γ(d+12
) the volume of the d− 1-dimensional unit sphere Sd−1.
Proof: (1) is fundamental. We directly see that
D±(s) = m− α2
2
(d− 1
d
)|Sd−1| (Hρ(s)∓ πiρ(s)) , (3.35)
where ρ(s) = |ϕ(√s)|2s d−2
2 and Hρ denotes the Hilbert transform of ρ, i.e.,
Hρ(s) = limε↓0
∫|s−x|>ε
ρ(x)
s− xdx.
38
Namely4
D±(s) = m− α2
2
(d− 1
d
)|Sd−1|
(limε↓0
∫|s−x|>ε
|ϕ(√x)|2|x| d−2
2
s− xdx∓ πi|ϕ(
√s)|2s
d−22
).
(3.36)
Then (2) and (3) follow from this. 2
Lemma 3.8 (1) Suppose (1),(2) and (3) of Assumption 3.4 . Then there exists
ε > 0 such that |D±(s)| > ε for s ∈ [0,∞).
(2) Suppose (1),(2’) and (3) of Assumption 3.4 . Then there exists ε > 0 such
that |D±(s)| > ε for s ∈ [λ2,Λ2] and D±(s) has at most one zero point on open
interval (Λ2,∞).
Proof: By Assumption 3.4 (2), the imaginary part of D±(s) does not vanish, and
Assumption 3.4 (2) implies that the real part of D± is also Lipshitz continuous with
the same order C as that of ρ(s) = |ϕ(√s)|2s d−2
2 , since the real part is the Hilbert
transformation of ρ. Then the real part of D±(s) goes to m > 0 as s → ∞ [Tit37,
p.145,5.15]. In particular, there exists ε > 0 such that sups∈[0,∞) |D±(s)| > ε.
Next Assumption 3.4 (2’) implies that the imaginary part of D+(s) 6= 0 for
s ∈ [λ2,Λ2]. The real part of D+(s) is m− α2
2
(d−1d
)|Sd−1|
∫ Λ2
λ2
|ϕ(√x)|2x
d−22
s−x dx. Then it
is monotonously increasing on [0, λ2) ∪ (Λ2,∞) with D+(0) = m + α2(d−1d
)‖ϕ/ω‖2
and lims→∞D+(s) = m. Then the lemma follows. 2
Define
Gf(k) = limε↓0
∫Rd
f(k′)
(ω(k)2 − ω(k′)2 + iε)(ω(k)ω(k′))d−2
2
dk′. (3.37)
It is seen that
Gf(k) =1
2|k| d−22
(HF (|k|2)− πi[f ](|k|)|k|
d−22
), (3.38)
4[HS01, (4.1)] is incorrect. ∓2πi|ϕ(√s)|2s(d−2)/2 is changed to ∓πi|ϕ(
√s)|2s(d−2)/2.
39
where HF denotes the Hilbert transform of F (x) = [f ](√x)x
d−24 and [f ](r) =∫
Sd−1f(r, v)dv and v is the volume element on Sd−1. Define the running effective
mass by
meff(k) = D+(ω(k)2) (3.39)
and we set
Tµνf = δµνf + αQωd−2
2 Gωd−2
2 dµνϕf, (3.40)
where
Q(k) =αϕ(k)
meff(k). (3.41)
The operator Tµν is then given by
Tµνf(k) = δµνf(k) + α2
∫dµν(k
′)ϕ(k)ϕ(k′)f(k′)
D+(|k|2)(|k|2 − |k′|2 + i0)dk′.
Remark 3.9 We give a comment on the definition of Q. Under (2) of Assumption
3.4, meff(k) 6= 0 for all k ∈ Rd, then Q is well defined. On the other hand under
(2’) of Assumption 3.4, meff(k) probably has one zero point in (Λ2,∞) Then Q is
well defined, since the support of the numerator is suppϕ = [λ,Λ]. Notice also that
meff(k) 6= 0 at least for λ− δ ≤ |k| ≤ Λ + δ with some δ > 0.
Example 3.10 (Running effective mass for sharp cutoff) Let us compute an
effective mass meff(k) with sharp cutoff. Let
ϕ(k) = 1l[λ,Λ](|k|) (3.42)
be the indicator function on λ ≤ |k| ≤ Λ. Suppose d = 3. Then
Hρ(s) = limε↓0
∫|s−x|>ε
1l[λ,Λ](√x)√x
s− xdx = lim
ε↓0
∫|s−x|>ε
1l[λ2,Λ2](x)√x
s− xdx.
Let s ∈ [0, λ). Then we see that
Hρ(s) =
∫ Λ2
λ2
√x
s− xdx =
∫ Λ
λ
2t2
s− t2dt = −2(Λ− λ)− 2s
∫ Λ
λ
1
t2 − sdt
= −2(Λ− λ)−√s
∫ Λ
λ
1
t−√s− 1
t+√sdt
= −2(Λ− λ) +√s log
((λ−
√s)(√s+ Λ)
(√s+ λ)(Λ−
√s)
).
40
Let s ∈ (Λ,∞). Then in a similar computation we have
Hρ(s) = −2(Λ− λ) +√s log
((√s− λ)(
√s+ Λ)
(√s+ λ)(
√s− Λ)
).
Finally let s ∈ [λ,Λ].∫|s−x|>ε
1l[λ,Λ](√x)√x
s− xdx =
(∫ s−ε
λ2
+
∫ Λ2
s+ε
) √x
s− xdx
=
(∫ √s−ελ
+
∫ Λ
√s+ε
)2t2
s− t2dt
= 2
(∫ √s−ελ
+
∫ Λ
√s+ε
)(−1 +
s
s− t2)dt
→ −2(Λ− λ) + limε↓0
2s
(∫ √s−ελ
+
∫ Λ
√s+ε
)1
s− t2ds
as ε→ 0. Directly we have
2s
(∫ √s−ελ
+
∫ Λ
√s+ε
)1
s− t2ds
=√s
(log
((√s+√s− ε)(
√s− λ)
(√s−√s− ε)(
√s+ λ)
)− log
((√s+√s+ ε)(Λ−
√s)
(√s+ ε−
√s)(√s+ Λ)
))=√s log
((√s− λ)(
√s+ Λ)
(√s+ λ)(Λ−
√s)
√s+√s− ε)(
√s+ ε−
√s)√
s−√s− ε)(
√s+ ε+
√s)
)→√s log
((√s− λ)(
√s+ Λ)
(√s+ λ)(Λ−
√s)
)as ε→ 0. Thus we obtain that
Hρ(s) = −2(Λ− λ) +√s log
∣∣∣∣(√s+ Λ)(√s− λ)
(√s+ λ)(
√s− Λ)
∣∣∣∣ . (3.43)
Since(d−1d
)|Sd−1| = 8π/3 for d = 3, the running effective mass with sharp cutoff
(3.42) is given by
meff(k) = m+8πα2
3(Λ− λ)− 4πα2
3
(|k| log
∣∣∣∣(|k|+ Λ)(|k| − λ)
(|k|+ λ)(|k| − Λ)
∣∣∣∣− iπ1l[λ,Λ](|k|)√|k|).
(3.44)
41
Lemma 3.11 The operator G is a bounded and antisymmetric operator on L2(Rd),
i.e., G∗ = −G.
Proof: Let f ∈ L2(Rd). The imaginary part of Gf is −π2[f ](|k|) and ‖[f ]‖ ≤
|Sd−1|‖f‖. The real part of Gf is given by the Hilbert transform; HF with F (x) =
[f ](√x)x
d−24 . The Hilbert transformation is bounded operator on L2(Rd) with
‖Hf‖ = π‖f‖ and then we have
‖HF‖2 =1
4
∫|HF (|k|2)|2
|k|d−2dk =
1
2
∫ ∞0
|HF (s)|2ds|Sd−1| =1
2|Sd−1|π2‖F‖2
=1
2|Sd−1|π2
∫[f ](√s)2s
d−22 ds = |Sd−1|π2
∫[f ](r)2rd−2dr
≤ |Sd−1|2π2
∫f(r, v)2rd−2drdv = |Sd−1|2π2‖f‖2.
Then the lemma follows. 2
The operator Tµν , functions Q and ϕ satisfy some algebraic relations. We list
up them in the lemma below, where we assume that m is not only positive but also
negative. Precisely we assume that
m > −(d− 1
d
)α2‖ϕ/ω‖2, m 6= 0 (3.45)
for mathematical generality. When 0 > m > −(d−1d
)α2‖ϕ/ω‖2, we see that D(s) ∈
R for s ≤ 0, lims→−∞
D(s) = m < 0 and D(0) = m+(d−1d
)α2‖ϕ/ω‖2 > 0. Then D(z)
has a unique zero point of order one in (−∞, 0). We denote the zero point by −E2
(E > 0), and γ is defined only in the case of m < 0 by
γ = D′(−E2)−1/2, (3.46)
where we can directly see that
D′(−E2) = α2
(d− 1
d
)∫|ϕ(k)|2
E2 + ω(k)2dk.
Let T ∗µν = (Tµν)∗, i.e.,
T ∗µνf = δµνf − αdµνϕωd−2
2 Gωd−2
2 Qf.
Note that G∗ = −G.
42
Lemma 3.12 (Algebraic relations) [Ara83-a, Ara83-b] Suppose Assumption 3.4
and (3.45). Let θ(m) =
1, m < 00, m > 0.
Set
ρµν(k) = δµν − dµν(k) = kµkν/|k|2, η =1(
d−1d
)|Sd−1|
, γ± = D+/D−,
Fµν = dµναϕ
E2 + ω2, [f ](|k|) =
∫Sd−1
f(|k|, v)dv.
Then the operator Tµν has the following properties:
1. ‖ωn/2Tµνf‖ ≤ C‖ωn/2f‖ for n = −1, 0, 1.
2. Tµνf = Tµν f , Tµν = Tνµ and if f is rotation invariant, then so is Tµνf .
3. T ∗µνdνaTabf = dµbf − θ(m)γ2(Fab, f)Faµ. In particular
erµT∗µνdνaTabe
sbf = δrsf − θ(m)γ2(esbFab, f)erµFaµ. (3.47)
4. TµνdνaT∗abf = dµbf + α(ρµbϕGQ−QGϕρµb)f . In particular
erµTµνdνaT∗abe
sbf = δrsf. (3.48)
5. T µνf = γ±Tµνf + (1− γ±) (δµνf − η[dµνf ]).
6. TµνdνaT∗abf = dµbf − (1− γ±)η[dµbf ] + α(ρµbϕGQ−QGϕρµb)f . In particular
erµTµνdνaT∗abe
sb = erµdµbe
sb − (1− γ±)ηerµ[dµbesbf ]. (3.49)
7. T ∗µνdνahTab = T ∗µνdνahTab for rotation invariant function h.
8. [ω2, Tµν ]f = α(dµνϕ, f)Q and [ω2, T ∗µν ]f = −α(Q, f)dµνϕ.
9. αTµνϕ = δµνmQ. In particular αerµTµνdνaϕ = δµamQ.
10. erµTµνT∗νae
saf = δrsf .
11.
(dνa
Q
ω,
1
ωTµνf
)=
α
meff
(dµa
ϕ
ω,f
ω
)− θ(m)
γ2
E2(Fµa, f).
43
12.
(dνa
Q√ωn, h
1
ωTµνf
)=
(dνa
Q√ωn, h
1
ωT µνf
)for rotation invariant function h
with n = 0, 1, 2.
13. erµTµνFνa = 0 (m < 0).
14. α(ϕ, Fµν) = −mδµν (m < 0).
15. T ∗µνdνaQ = θ(m)γ2Fµa.
16. (Fµa, Faν) = δµν1
γ2(m < 0).
Statements (3)-(7) are used in Lemma 3.14 to show some symplectic structure, (8)
and (9) in Lemma 3.21 to show some commutation relations, (10) in the proof of
Theorem 3.6, (11) and (12) in Lemma 3.17 and the proof of Theorem 3.6, and
(13)-(16) in Section 3.10.
Proof of Lemma 3.12:
Note that for rotation invariant functions f and g,
(dµνf, g) = δµν
(d− 1
d
)(f, g)
and the identity
α2
2
(d− 1
d
)|Sd−1|
ϕ(√s)2s
d−22
|D+(s)|2=
1
2πi
(1
D−(s)− 1
D+(s)
)(3.50)
holds5 by D+(s) − D−(s) = πiα2(d−1d
)|Sd−1|ϕ(
√s)2s
d−22 . We set G = ω
d−22 Gω
d−22
for the notational convenience.
1. By (3) of Assumption 3.4 we can see that
‖√ωTµνf‖ ≤ ‖
√ωδµνf‖+ |α|‖ 1
D+(ω2(·))‖∞‖ω
d−12 ϕ‖∞‖ω
d−32 ϕ‖∞‖G‖2‖
√ωf‖,
‖ 1√ωTµνf‖ ≤ ‖
1√ωδµνf‖+ |α|‖ 1
D+(ω2(·))‖∞‖ω
d−12 ϕ‖∞‖ω
d−32 ϕ‖∞‖G‖2‖
1√ωf‖,
‖Tµνf‖ ≤ ‖δµνf‖+ |α|‖ 1
D+(ω2(·))‖∞‖ω
d−22 ϕ‖∞‖ω
d−22 ϕ‖∞‖G‖2‖f‖.
5When ϕ(k) = 0 for |k| > Λ or |k| < λ ((2’) of Assumption 3.4), (3.50) is valid for s ∈ [λ2,Λ2].
44
Then 1. follows6.
2. This follows from the definition of Tµν .
3. We have
(T ∗µνdνaTabf, g)
= (dνaTabf, Tµνg)
= (dµbf, g) + α(f, dνbQGdµνϕg) + α(daµQGdabϕf, g)
+ α2
(d− 1
d
)(QGdabϕf, QGdaµϕg)
= I + II + III + IV.
We compute IV as
IV = limt↓0
α2
∫ (d−1d
)|Q(k)|2F (k′, k′′)
(|k|2 − |k′|2 − it)(|k|2 − |k′′|2 + it)dkdk′dk′′
= limt↓0
α2
2
∫ (∫ ∞0
(d−1d
)ϕ2(√s)s
d−22 |Sd−1|F (k′, k′′)
(s− |k′|2 − it)(s− |k′′|2 + it)|D+(s)|2ds
)dk′dk′′
= limt↓0
1
2πi
∫ (∫ ∞0
F (k′, k′′)
(s− |k′|2 − it)(s− |k′′|2 + it)
(1
D−(s)− 1
D+(s)
)ds
)dk′dk′′,
where F (k′, k′′) = dab(k′)daµ(k′′)ϕ(k′)ϕ(k′′)f(k′)g(k′′). By a contour integral
on the cut plane Cε,δ,R (Figure 2), we have
IV = limt↓0
limε,δ↓0R→∞
1
2πi
∫ (∫Cε,δ,R
−F (k′, k′′)
(z − |k′|2 − it)(z − |k′|2 + it)D(z)dz
)dk′dk′′
= limt↓0
∫−α2F (k′, k′′)
D(|k′′|2 − it)(|k′′|2 − |k′|2 − 2it)dk′dk′′
+ limt↓0
∫−α2F (k′, k′′)
D(|k′|2 + it)(|k′|2 − |k′′|2 + 2it)dk′dk′′
+ limt↓0
∫−α2F (k′, k′′)γ2θ(m)
(E2 + |k′|2 + it)(E2 + |k′′|2 − it)dk′dk′′
= −α2
(f, dab
ϕ
D+
Gdaµϕg
)− α2
(daµ
ϕ
D+
Gdabϕf, g
)− θ(m)γ2(f, Fab)(Faµ, g).
6When ϕ(k) = 0 for |k| > Λ or |k| < λ ((2’) of Assumption 3.4), it is understood that‖ 1D+(ω2(·))‖∞ = supλ≤|k|≤Λ | 1
D+(ω2(k)) |.
45
Figure 2: Cut plane Cε,δ,R
Then IV = −II − III − θ(m)γ2(f, Fab)(Faµ, g). Hence the desired result is
obtained.
4. We see that
(TµνdνaT∗abf, g)
= (dµbf, g)− α(dµbϕGQf, g)− α(f, dµbϕGQg) + α2(dµbϕGQf, ϕGQg)
= I + II + III + IV.
We have
IV = limt↓0
α2δµb
∫ (d−1d
)ϕ(k)2H(k′, k′′)
(|k|2 − |k′|2 − it)(|k|2 − |k′′|2 + it)dkdk′dk′′,
46
where H(k′, k′′) = Q(k′)Q(k′′)f(k′)g(k′′). We can see that
IV = limt↓0
α2
2δµb
∫ (∫ ∞0
(d−1d
)ϕ(√s)2s
d−22 |Sd−1|H(k′, k′′)
(s− |k′|2 − it)(s− |k′′|2 + it)ds
)dk′dk′′
= limt↓0
α2
2δµb
∫ (∫ ∞0
|D+(s)|2(d−1d
)ϕ(√s)2s
d−22 |Sd−1|H(k′, k′′)
(s− |k′|2 − it)(s− |k′′|2 + it)|D+(s)|2ds
)dk′dk′′
= limt↓0
1
2πiδµb
∫ (∫ ∞0
|D+(s)|2H(k′, k′′)
(s− |k′|2 − it)(s− |k′′|2 + it)
×(
1
D−(s)− 1
D+(s)
)ds
)dk′dk′′
= limt↓0
1
2πiδµb
∫ (∫ ∞0
(D+(s)−D−(s))H(k′, k′′)
(s− |k′|2 − it)(s− |k′′|2 + it)ds
)dk′dk′′.
It can be computed by a contour integral on the cut plane Cε,δ,R as
IV = limt↓0
limε,δ↓0R→∞
1
2πiδµb
∫ (∫Cε,δ,R
D(z)H(k′, k′′)
(z − |k′|2 − it)(z − |k′′|2 + it)dz
)dk′dk′′
= limt↓0
δµb
∫D(|k′|2 + it)−D(|k′′|2 − it)
|k′|2 − |k′′|2 + 2itH(k′, k′′)dk′dk′′
= αδµb(f, ϕGQg) + αδµb(ϕGQf, g).
Then the desired result is obtained.
5. We have T µνf(k) = δµνf(k) + limt↓0
αQ(k)
∫dµν(k
′)ϕ(k′)f(k′)
|k|2 − |k′|2 − itdk′, and
∫dµν(k
′)ϕ(k′)f(k′)
|k|2 − |k′|2 − itdk′
=
∫dµν(k
′)ϕ(k′)f(k′)
|k|2 − |k′|2 + itdk′ + 2i
∫tdµν(k
′)ϕ(k′)f(k′)
(|k|2 − |k′|2)2 + t2dk′.
Since π−1∫
tx2+t2
f(x)dx→ f(0) as t→ 0, we see that
2i
∫tdµν(k
′)ϕ(k′)f(k′)
(|k|2 − |k′|2)2 + t2dk′ = i
∫ ∞0
t[dµνf ](√s)ϕ(√s)s
d−22
(|k|2 − s)2 + t2ds
→ πi[dµνf ](ω)ϕ(ω)ωd−2.
47
Then
T µνf = δµνf + αQ(Gdµνϕf + πiωd−2[dµνf ](ω)ϕ
)Notice that Q = γ±Q. Hence
Tµνf = δµνf + αγ±Q(Gdµνϕf + πiωd−2ϕ[dµνf ](ω)
)= γ±Tµνf + δµν(1− γ±)f + πiαγ±Qϕ[dµνf ](ω)ωd−2.
Since D+(s)−D−(s) = πiα2(d−1d
)ϕ2(√s)s
d−22 |Sd−1|, we see that
πiαωd−2γ±Qϕ = πiαωd−2αϕ2
D−(ω2)= η
D+(ω2)−D−(ω2)
D−(ω2)= η(γ± − 1).
Then the desired result is obtained.
6. Note that [dµν ] = δµν/η.
TµνdνaT∗abf = γ±TµνdνaT
∗abf + (1− γ±)(δµνdνaT
∗abf − η[dµνdνaT
∗abf ])
Notice that
δµνdνaT∗abf − η[dµνdνaT
∗abf ] = dµbf − η[dµbf ] + αρµbϕGQf.
Thus
TµνdνaT∗abf
= dµbf − (1− γ±)η[dµbf ] + α(1− γ±)ρµbϕGQf + αγ±(ρµbϕGQ−QGϕρµb)= dµbf − (1− γ±)η[dµbf ] + αρµbϕGQf − αQGϕρµbf.
Since erµρµb = 0, we in particular obtain (3.49).
7. We see that
(T ∗µνdνahTabf, g) =(dνahTabf, Tµνg)
=(γ±dνahTabf, γ±Tµνg)
+ (dνah(1− γ±)(δabf − η[dabf ]), γ±Tµνg)
+ (dνahγ±Tabf, (1− γ±)(δµνg − η[dµνg]))
+ (dνah(1− γ±)(δabf − η[dabf ]), (1− γ±)(δµνg − η[dµνg]))
=I + II + III + IV.
48
We have I = (dνahTabf, Tµνg). We will show that II + III + IV = 0. Since h
and γ± are rotation invariant, we have
II =
∫h(1− γ±)γ±([dµbf g]− η[dµag][dabf ])dk
III =
∫h(1− γ±)γ±([dµbf g]− η[dµνg][dνbf ])dk
IV =
∫h|1− γ±|2
× ([dµbf g]− η[dµag][dabf ]− η[dµνg][dνbf ] + η2[dνa][dµνg][dabf ])dk.
Notice that
(1− γ±)γ± = γ± − 1,
(1− γ±)γ± = γ± − 1,
|1− γ±|2 + γ± − 1 + γ± − 1 = 0,
− η[dµνg][dνbf ] + η2[dνa][dµνg][dabf ] = −η[dµνg][dνbf ] + ηδνa[dµνg][dabf ] = 0.
Then II + III + IV = 0 follows.
8. We see that
[ω2, Tµν ]f = limt↓0
α
∫(|k|2 − |k′|2)Q(k)dµν(k
′)ϕ(k′)f(k′)
|k|2 − |k′|2 + itdk′ = α(dµνϕ, f)Q.
9. We see that
αTµνϕ = αδµνϕ+ α2 ϕ
D+
Gdµνϕ2 = αδµνϕ
(1 +
α2(d−1d
)Gϕ2
m− α2(d−1d
)Gϕ2
)= mδµνQ.
Here we used that D+ = m− α2(d−1d
)Gϕ2. In particular it follows that
αerµTµνdνaϕ = αerµTµaϕ− erµρµaϕ = αerµTµaϕ = meraQ.
10. This is shown in the same way as 4.
11. We have(dνa
Q
ω,
1
ωTµνf
)=
(dµa
Q
ω,f
ω
)+
(dνa
Q
ω,αQ
ωGdµνϕf
)= I + II.
49
By (3.50) we have
II = limt↓0
α
∫|Q(k)|2dµν(k′)dνa(k)ϕ(k′)f(k′)
(|k|2 − |k′|2 + it)|k|2dkdk′
= limt↓0
α3
2
∫ (∫ ∞0
(d−1d
)ϕ2(√s)s
d−22 dµa(k
′)ϕ(k′)f(k′)|Sd−1|(s− |k′|2 + it)s|D+(s)|2
ds
)dk′
= limt↓0
α
2πi
∫ (∫ ∞0
1
(s− |k′|2 + it)s
(1
D−(s)− 1
D+(s)
)F (k′)ds
)dk′,
where F (k′) = dνa(k′)ϕ(k′)f(k′). By a contour integral on the cut plane Cε,δ,R,
we have
1
2πi
∫1
(s− |k′|2 + it)s
(1
D−(s)− 1
D+(s)
)ds
= − 1
2πilimε,δ↓0R→∞
∫Cε,δ,R
1
(z − |k′|2 + it)zD(z)dz
= − 1
(|k′|2 − it)D(|k′|2 − it)− θ(m)γ2
(E2 + |k′|2 + it)E2+
1
meff(|k′|2 − it).
Then
II
= limt↓0
∫ αF (k′)
meff(|k′|2 − it)− αF (k′)
(|k′|2 − it)D(|k′|2 − it)− αθ(m)γ2F (k′)
(E2 + |k′|2 + it)E2
dk′
=α
meff
(dµa
ϕ
ω,f
ω
)−(dµa
Q
ω,f
ω
)− θ(m)
γ2
E2(Fµa, f) .
Hence we have
I + II =α
meff
(dµa
ϕ
ω,f
ω
)− θ(m)
γ2
E2(Fµa, f) .
12. Note that∫h(k)dµν(k)f(k)dk = η
∫h(k)dµa(k)[daνf ](|k|)dk for rotation in-
variant function h. From 5. it follows that(dνa
Q√ωn, h
1
ωTµνf
)=
(γ±dνa
Q√ωn, γ±
1
ωhTµνf
)+
(γ±dνa
Q√ωn, h
1
ω(1− γ±)(δµνf − η[dµνf ])
).
50
Since γ±dνaQ√ωnh 1√
ω(1 − γ±) is rotation invariant, the second term vanishes.
Then the claim is proven.
13. We have
(erµTµνFνa, g) = (Fµa, erµg)− (Fνa, αdµνϕGQe
rµg) = I− II.
Then
II = α2
∫ (d−1d
)ϕ(k)2Q(k′)era(k
′)g(k′)
(|k|2 + E2)(|k|2 − |k′|2 + it)dkdk′.
We see that∫ϕ(k)2
(|k|2 + E2)(|k|2 − |k′|2 + it)dk
=
∫ (ϕ(k)2
|k|2 − |k′|2 + it− ϕ(k)2
|k|2 + E2
)1
E2 + |k′|2 − itdk.
By the definitions of D− and −E2, we have
limt↓0
∫α2(d−1d
)ϕ(k)2
|k|2 − |k′|2 + itdk = D−(|k′|2)−m, (3.51)∫
α2(d−1d
)ϕ(k)2
|k|2 + E2dk = D−(−E2)−m = −m. (3.52)
Thus we get
II =
∫ϕ(k′)era(k
′)g(k′)
E2 + |k′|2dk′ = (Fba, e
rbg)
and I− II = 0 follows.
14. We see that α(ϕ, Fµν) = α2δµν
(d− 1
d
)∫ϕ(k)2
E2 + |k|2dk = −mδµν by (3.52).
15. We see that
(T ∗µνdνaQ, f) = (dνaQ, δµνf) + (dνaQ,αQGdµνϕf) = I + II.
51
We have
II = limt↓0
α
∫|Q(k)|2dµν(k′)dνa(k)ϕ(k′)f(k′)
|k|2 − |k′|2 + itdkdk′
= limt↓0
α3
2
∫ (∫ ∞0
(d−1d
)ϕ(√s)2s
d−22 dµa(k
′)ϕ(k′)f(k′)|Sd−1|(s− |k′|2 + it)|D+(s)|2
ds
)dk′
= limt↓0
α
2πi
∫ (∫ ∞0
1
s− |k′|2 + it
(1
D−(s)− 1
D+(s)
)F (k′)ds
)dk′,
where F (k′) = dµa(k′)ϕ(k′)f(k′). By a contour integration on the cut plane
Cε,δ,R, we compute as
1
2πi
∫ ∞0
1
s− |k′|2 + it
(1
D−(s)− 1
D+(s)
)ds
= − limε,δ↓0R→∞
1
2πi
∫Cε,δ,R
1
(z − |k′|2 + it)D(z)dz
= − 1
D(|k′|2 − it)+
γ2
E2 + |k′|2 + it.
Then we have
II = limt↓0
α
∫ (− 1
D(|k′|2 − it)+
γ2
E2 + |k′|2 + it
)F (k′)dk′
= −(dµaQ, f) + γ2 (Fµa, f) .
Hence I + II = γ2(Fµa, f) follows.
16. We have
(Fµa, Faν) = δµνα2
(d− 1
d
)∫ϕ(k)2
(|k|2 + E2)2dk
= δµν
(d− 1
d
)α2
2
∫ ∞0
ϕ(√s)2s
d−22
(s+ E2)2ds
=1
2πiδµν
∫ ∞0
D+(s)−D−(s)
(s+ E2)2ds.
Hence by a contour integral on the cut plane Cε,δ,R, we have
(Fµa, Faν) = limε,δ↓0R→∞
1
2πiδµν
∫Cε,δ,R
D(z)
(z + E2)2dz = δµνD
′(−E2).
52
Then the proof is complete. 2
3.4.2 Intertwining operator
Now we introduce the class of functions. Let
Mn = f |ωnf ∈ L2(Rd). (3.53)
Let Aµ(f) = Aµ(f) and Πµ(g) = Πµ(g), i.e.,
Aµ(f) =1√2
∫1√ω(k)
ejµ(k) (a∗(k, j)f(k) + a(k, j)f(−k)) dk,
Πµ(f) =i√2
∫ √ω(k)ejµ(k) (a∗(k, j)f(k)− a(k, j)f(−k)) dk.
Then [Aµ(f), Πν(g)] = i(dµν f , g) holds, and
[Aµ, Πν(g)] = i(dµν ˆϕ, ˆg) = i(dµν˜ϕ, g) = i(dµν ¯ϕ, g) = i(dµνϕ, g),
where we used ¯ϕ = ϕ. Then we define
Bp(f, j) =1√2
Aµ(T ∗µν√ωejνf
)+ iΠµ
(T ∗µν
1√ωejνf
)−(p · ejQ
ω,f√ω
), (3.54)
B∗p(f, j) =1√2
Aµ
(T∗µν
√ωT ejνf
)− iΠµ
(T∗µν
1√ω
T ejνf
)−(p · ejQ
ω,f√ω
)(3.55)
for f ∈M0 ∩M−1/2, where T f(k) = f(k) = f(−k).
Remark 3.13 Note that condition f ∈M−1/2 is not needed for the definition of B#p
for p = 0.
We have
Bp(f, j) = a(W+ijf, i) + a∗(W−ijf, i)−(p · ej Q√
2ω,f√ω
), (3.56)
B∗p(f, j) = a(W−ijf, i) + a∗(W+ijf, i)−(p · ej Q√
2ω,f√ω
)(3.57)
53
for f ∈M0 ∩M−1/2, where
W+ij =1
2eiµ
(1√ωT ∗µν√ω +√ωT ∗µν
1√ω
)T ejν , (3.58)
W−ij =1
2eiµ
(1√ωT ∗µν√ω −√ωT ∗µν
1√ω
)ejν . (3.59)
Let W± = (W±ij)1≤i,j≤d−1 : ⊕d−1L2(Rd)→ ⊕d−1L2(Rd) and
W =
(W+ W−W− W+
):
2⊕(⊕d−1L2(Rd))→
2⊕(⊕d−1L2(Rd)).
We set
bW (f, j) = a(W+ijf, i) + a∗(W−ijf, i), (3.60)
b∗W (f, j) = a(W−ijf, i) + a∗(W+ijf, i). (3.61)
for f ∈M0.
Lemma 3.14 (Symplectic structure) Suppose Assumption 3.4 and that m >
−(d−1d
)α2‖ϕ/ω‖2, m 6= 0. Then
W ∗+W+ −W ∗
−W− = 1l, (3.62)
W∗+W− −W
∗−W+ = 0, (3.63)
W+W∗+ −W−W
∗− = 1l + θ(m)Z+, (3.64)
W−W∗+ −W+W
∗− = θ(m)Z−, (3.65)
where
θ(m) =
1 m < 0,0 m > 0,
Z±,ijf = ∓1
2γ2
(√ωF i
µ
(F jµ√ω, f
)± 1√
ωF iµ(√ωF j
µ, f)
),
F jµ =
αϕ
E2 + ω2ejµ,
γ = D′(−E2)−1/2 =
((d− 1
d
)∫α2ϕ(k)2
E2 + ω(k)2dk
)−1/2
.
54
In particular in the case of m > 0,
W ∗+W+ −W ∗
−W− = 1l, (3.66)
W∗+W− −W
∗−W+ = 0, (3.67)
W+W∗+ −W−W
∗− = 1l, (3.68)
W−W∗+ −W+W
∗− = 0, (3.69)
holds, i.e., W ∈ Sp.
Proof: These relations are proven by making use of algebraic relations stated in
Lemma 3.12. By (3.48) in Lemma 3.12 we see that
W ∗+jkW+kj′ −W ∗
−jkW−kj′
=1
2
(√ωejνTµνdµaT
∗abe
j′
b
1√ω
+1√ωejνTµνdµaT
∗abe
j′
b
√ω
)= δjj′1l.
Then (3.62) follows. By (3.49) of Lemma 3.12 we see that
erµ√ωTµνdνaT
∗ab
1√ωesb = erµ
1√ωTµνdνaT
∗ab
√ωesb.
Then
W ∗+jkW
∗−kj′ −W ∗
+jkW−kj′
=1
2
(√ωejνTµνdµaωT
∗ab
1√ω
T ej′
b −√ωejνT
∗µνdµaTabT ej
′
b
1√ω
)= 0.
Then (3.62) follows. By (7) of Lemma 3.12 we have
W+jkW∗+kj′ −W ∗
−jkW∗−kj′ =
1
2
(√ωejνT
∗µνdµaTabe
j′
b
1√ω
+1√ωejνT
∗µνdµaTabe
j′
b
√ω
).
By (3.47) yields that
W+jkW∗+kj′ −W ∗
−jkW∗−kj′ = δjj′1l + θ(m)Z+jj′ .
Then (3.62) follows. By (7) of Lemma 3.12 we have
W−jkW∗−kj′ −W ∗
+jkW∗−kj′
=1
2
(ejµ
1√ωT ∗µνdνaTab
√ωT ej
′
b − ejµ
√ωT ∗µνdνaTab
1√ω
T ej′
b
).
55
Then from (3.47) it follows that
W−jkW∗−kj′ −W ∗
+jkW∗−kj′ = θ(m)Z−ij.
Then (3.62) follows. 2
Lemma 3.15 Suppose Assumption 3.4 and m > 0. Then W ∈ Sp2, i.e., W− ∈ I2.
Proof: It is enough to show that W−ij is a Hilbert-Schmidt operator on L2(Rd) for
each i, j. By the definition of W−, we can see that W−ij is the integral operator with
the integral kernel:
W−ij(k, k′) =
α2
2
ϕ(k)ϕ(k′)eiµ(k)ejµ(k′)√|k|√|k′|(|k|+ |k′|)D−(|k′|2)
.
Since
|W−ij(k, k′)| ≤ C(α2/2)(ϕ(k)/|k|)(ϕ(k′)/|k′|)
with some constant C by the assumption ϕ/ω ∈ L2(Rd), W−ij(·, ·) ∈ L2(Rd × Rd).
Then W−ij is a Hilbert-Schmidt operator. 2
Let m > 0. By the general result obtained in Section 2.1 we can see that
canonical commutation relations hold:
[Bp(f, j), B∗p(g, j
′)] = δjj′(f , g),
[B∗p(f, j), B∗p(g, j
′)] = 0,
[Bp(f, j), Bp(g, j′)] = 0,
and the adjoint relation (F,Bp(f, j)G) = (B∗p(f , j)F,G) is satisfied. Furthermore
for W =
(W+ W−W− W+
)we have
a(f, j) = bW (W ∗+ijf, i)− b∗W (W ∗
−ijf, i), (3.70)
a∗(f, j) = −bW (W ∗−ijf, i) + b∗W (W ∗
+ijf, i) (3.71)
for f ∈ L2(Rd).
56
Lemma 3.16 (Intertwining operator) Let m > 0. Then the intertwining opera-
tor UW associated with W =
(W+ W−W− W+
)∈ Sp2 is given by
UW = C exp
(−1
2∆∗W−W
−1+
):exp
(−N
1l−(W−1+ )∗
): exp
(−1
2∆−W−1
+ W−
)(3.72)
and the normalizing constant C by C = det(1l− (W−W−+ )∗(W−W
−+ ))1/4.
Proof: This follows from Proposition 2.5. 2
3.4.3 Displacement operator
We will construct the displacement operator associated with W =
(W+ W−W− W+
)and
the vector
L = −pµ
e1µ
Q√ω3
...
...
ed−1µ
Q√ω3
∈ ⊕d−1L2(Rd). (3.73)
Lemma 3.17 (Displacement operator) Suppose that
∫ϕ2
ω3dk < ∞. Then the
displacement operator associated with W =
(W+ W−W− W+
)and L is given by
Sp = exp(−iΠp) (3.74)
with the generator
Πp =i√2
α
meff
a∗(
p · ejϕω3/2
, j)− a(p · ejϕω3/2
, j)
. (3.75)
Proof: The generator of the displacement operator is given by
Πp = − i√2
(bW (p · ejQω3/2
, j)− b∗W (p · ejQω3/2
, j)).
57
We compute the right hand side above. Then Πp = i√2(a∗(ξj, j)− a(ξj, j)) with
ξi = W+ijp · ejQω3/2
−W−ijp · ejQω3/2
.
By (7), (9) and (12) of Lemma 3.12, we can see that T ∗µνdνaQω
= T ∗µνdνaQω
and
T ∗µνdνaQω2 = T ∗µνdνa
Qω2 , and we have
ξj = ejµ√ωT ∗µνdνapa
Q
ω2=
α
meff
p · ej ϕ
ω3/2
by (11) of Lemma 3.12 under the condition m > 0. 2
Definition 3.18 (Bogoliubov transformation) Let W =
(W+ W−W− W+
)and p ∈
Rd. Suppose
∫ϕ2
ω3dk <∞. Then we define the unitary operator Up by
Up = SpUW . (3.76)
Remark 3.19 In the case of p = 0, we do not need to assume that
∫ϕ2
ω3dk <∞ in
the definition of Up in (3.76).
Lemma 3.20 Suppose Assumption 3.4. (1) Let p = 0. Then U0 maps D(Hf) onto
itself and
U −10 B]
0(f, j)U0 = a](f, j). (3.77)
(2) In addition to Assumption 3.4, suppose that
∫ϕ2
ω3dk <∞. Then for all p ∈ Rd,
Up maps D(Hf) onto itself and
U −1p B]
p(f, j)Up = a](f, j). (3.78)
Proof: This follows from the general results of Theorem 2.6. 2
58
3.5 Diagonalization and time evolution of radiation fields
3.5.1 Diagonalization
In this section we diagonalize Hp by a unitary operator.
Lemma 3.21 Suppose Assumption 3.4. Let f ∈ M1 ∩M1/2 ∩M0 ∩M−1/2. Then
for all p ∈ Rd,
[Hp, Bp(f, j)] = −Bp(ωf, j), (3.79)
[Hp, B∗p(f, j)] = B∗p(ωf, j). (3.80)
Proof: By the algebraic relations in Lemma 3.12 we will check commutation relations,
[Aµ, B](f, j)] and [Hf , B
](f, j)]. By (9) of Lemma 3.12 we see that
[Aµ, Bp(f, j)]
= − 1√2
(dµaϕ, T∗ab
1√ωejbf) = − 1√
2(ejbTabdµaϕ,
1√ωf) = − 1√
2
m
α(ejµQ,
1√ωf).
By taking the adjoint we also obtain
[Aµ, B∗p(f, j)] =
1√2
m
α(ejµQ,
1√ωf).
Next we see that by the definition of Bp(f, j),
[Hf , Bp(f, j)] =1√2
[Hf , Aµ(T ∗µν√ωejνf) + iΠµ(T ∗µν
1√ωejνf)]
=1√2
−iΠµ(T ∗µν
√ωejνf)− Aµ(ω2T ∗µν
1√ωejνf)
.
By (8) of Lemma 3.12 we have
[Hf , Bp(f, j)] =−1√
2
iΠµ(T ∗µν
1√ωejνωf) + Aµ(T ∗µν
√ωejνωf)
+
α√2
(ejµQ,1√ωf)Aµ.
Together with them we can see that
[Hp, Bp(f, j)]
=1
m(pµ − αAµ)(−α)[Aµ, Bp(f, j)] + [Hf , Bp(f, j)]
= − 1√2
iΠµ(T ∗µν
1√ωejνωf) + Aµ(T ∗µν
√ωejνωf)
+
1√2
(p · ejQω
,ωf√ω
)
= −Bp(ωf, j).
59
Then (3.79) follows. (3.80) is similarly proven. 2
Lemma 3.22 Suppose Assumption 3.4. Let f ∈M0 ∩M−1/2. Then for all p ∈ Rd,
it follows that
eitHpB∗p(f, j)e−itHp = B∗p(e
itωf, j), (3.81)
eitHpBp(f, j)e−itHp = Bp(e
−itωf, j). (3.82)
Proof: Fix j and f . Let A = B∗p(f , j)+Bp(f, j) and Π = i(B∗p(f , j)−Bp(f, j)). Then
A and Π are essentially self-adjoint. We denote the self-adjoint extensions by the
same symbols. Let At = B∗p(eitωf , j)+Bp(e
−itωf, j) and At = eitHpAe−itHp . Then for
Φ ∈ Ffin we can see that ddtAtΦ = [Hp, A]Φ by Lemma 3.21, and d
dtAtΦ = [Hp, A]Φ.
Thus the function F (t) = (Φ, (At − At)Ψ) Φ,Ψ ∈ Ffin, satisfies that ddtF (t) = 0,
and hence F (t) = F (0) = 0 for all t. Thus At = At on Ffin. By a limiting argument
At = At follows. Similarly we can see that eitHpΠe−itHpi(B∗p(eitωf, j)−Bp(e
itωf, j)).
Thus the lemma follows. 2
Set Ωp = UpΩ.
Lemma 3.23 Suppose Assumption 3.4. Then (1) and (2) follow.
(1) Ω0 ∈ σp(H0).
(2) Suppose that
∫ϕ2
ω3dk <∞. Then Ωp ∈ σp(Hp) for all p ∈ Rd.
Proof: We prove (2). Statement (1) is similarly proven. Since
Bp(f, j)Φ = Upa(f, j)U −1p Φ,
Bp(f, j)Φ = 0 for all f ∈ M0 ∩M−1/2 implies that Φ = aΩp with some a ∈ C. By
Lemma 3.22 we see that
Bp(f, j)e−itHpΩp = e−itHpBp(e
−itωf, j)Ωp = 0
for all f ∈M0 ∩M−1/2. Thus e−itHpΩp = at(p)Ωp with some at ∈ C. By the unitary
properties of eitHp we see that at(p) can be represented at(p) = eitEp with some
Ep ∈ R. Thus HpΩp = EpΩp follows. 2
60
Proof of Theorem 3.5
Proof: Let M = L.H∏n
i=1B∗p(fi, ji)Ωp|fi ∈ M0 ∩M−1/2, 1 ≤ ji ≤ d − 1, n ≥ 0.
Since a] leaves invariant, M is dense in F . Let Φ ∈M . Then we have
eitHpn∏i=1
B∗p(fi, ji)Ωp =n∏i=1
B∗p(eitωfi, ji)e
itEpΩp = Up
n∏i=1
a∗(eitωfi, ji)eitEpΩ
= Upeit(Hf+Ep)
n∏i=1
a∗(fi, ji)Ω = Upeit(Hf+Ep)U −1
p Φ.
Hence the theorem follows on M . By a limiting argument the theorem is proven.
2
We define the unitary operator on H ∼=∫ ⊕Rd Fdx by
U =
∫ ⊕Rd
Upeiπ
2Ndp. (3.83)
Proof of Theorem 3.6:
Proof: Let V ∈ L∞(Rd). By Theorem 3.5, we have
U −1HU = Heff +Hf + U −1VU − V + g (3.84)
on a core of the right-hand side above, e.g., C∞0 (Rd) ⊗alg [Ffin ∩ D(Hf)]. Since
H is self-adjoint on D(−∆) ∩ D(Hf), a limiting argument tells us that U maps
D(−∆) ∩D(Hf) onto itself and (3.84) is valid on D(−∆) ∩D(Hf). We see that
U −1e−ikxU = e−ikxe−iπ2NU −1
W exp
(−i α
meff
k · Π)
UW eiπ
2N ,
61
where Πµ = i√2
a∗(
ejµϕ
ω3/2 , j)− a(ejµϕ
ω3/2 , j)
. We have
U −1W ΠµUW
=i√2U −1W
a∗(ejµ
ϕ
ω3/2, j)− a(ejµ
ϕ
ω3/2, j)
UW
=i√2U −1W
b∗W
(W∗+ijejµ
ϕ
ω3/2+W
∗−ije
jµ
ϕ
ω3/2, i
)− bW
(W ∗−ije
jµ
ϕ
ω3/2+W ∗
+ijejµ
ϕ
ω3/2, i
)UW
=i√2
a∗(W∗+ijejµ
ϕ
ω3/2+W
∗−ije
jµ
ϕ
ω3/2, i
)− a
(W ∗−ije
jµ
ϕ
ω3/2+W ∗
+ijejµ
ϕ
ω3/2, i
)=
i√2
a∗(ω1/2eiaTabdbµ
ϕ
ω2, i
)− a
(ω1/2eiaTabdbµ
ϕ
ω2, i
).
By (10) and (11) of Lemma 3.12 we see that
ω1/2eiaTabdbµϕ
ω2=meff
αω1/2eiaTabT
∗bνdνµ
Q
ω2=meff
αω1/2eiaTabT
∗bνe
jνejµ
Q
ω2=meff
αeiµ
Q
ω3/2.
Since
e−iπ2N i a∗(g, j)− a(g, j) ei
π2N = a∗(g, j) + a(g, j), (3.85)
we have
e−iπ2NU −1
W ΠµUW eiπ
2N =
meff
α
1√2
a∗(ejµ
Q
ω3/2, j) + a(ejµ
Q
ω3/2, j)
=meff
αKµ.
Hence
U −1e−ikxU = e−ikxe−ik·K = T−1e−ikxT.
Let ρ ∈ C∞0 (Rd) be such that ρ(x) ≥ 0, suppρ ⊂ x ∈ Rd‖x| ≤ 1 and∫Rd ρ(x)dx =
1. Define ρε(x) = ρ(x/ε)/εd, ε > 0, and Vε = ρε ∗ V . We see that
U −1ρU = (2π)−d/2∫Rdρ(k)U −1e−ikxU dk =
∫Rdρ(k)T−1e−ikxTdk = T−1ρT.
Thus U −1VεU Ψ = T−1VεTΨ. By a limiting argument, we obtain (3.29) on D(−∆)∩D(Hf). 2
62
3.5.2 Time evolution of quantized radiation field
Now we can construct the solution to formal equation:
(∂2
∂t2−∆)Aµ(x, t) =
α
m(pν − Aν(x, t))ρνµ(x).
Theorem 3.24 (Time evolution of A) [Ara83-a, Ara83-b] Suppose Assumption
3.4 and that f is real-valued and f ∈M0 ∩M−1/2. Then for all p ∈ Rd,
eitHpAµ(f)e−itHp
=1√2
B∗p(e
itωejν1√ωTνµf , j) +Bp(e
−itωejν1√ωTνµ
˜f, j)
+
α
meff
pν(dµνϕ
ω,f
ω).
Proof: By the symplectic structure, W ∈ Sp, a] can be represented7 in terms of B]p:
a(f, j) = −B∗p(W ∗−ijf, i) +Bp(W
∗+ijf, i) +
α
meff
(p · ejϕ√
2ω,f√ω
), (3.86)
a∗(f, j) = B∗p(W∗+ijf, i)−Bp(W
∗−ijf, i) +
α
meff
(p · ejϕ√
2ω,f
ω
). (3.87)
Here we used (2.47) and (2.48) under S = W+, T = W− and L = −p · ejQ/ω3/2 and
that
−W+ij(−p · ejQ
ω3/2) +W−ij(−p · ej
Q
ω3/2) = eiµ
√ωT ∗µν
1√ωejνp · ej
Q
ω3/2=
α
meff
p · ei ϕ
ω3/2.
The quantized radiation field Aµ can be represented in terms of B#p . Inserting (3.86)
and (3.87) into Aµ we can see that
Aµ(f)
=1√2
B∗p(W
∗+ijR
jµ −W ∗
−ijRjµ, i) +Bp(W
∗+ijR
jµ −W ∗
−ijRjµ, i)
+
α
meff
pν(dµνϕ
ω,f
ω),
7[HS01, (4.7) and (4.8)] is incorrect. − αmeff
(p·ej ϕ√3ω3/2
, f) is changed to + αmeff
(p·ej ϕ√3ω3/2
, f) and
− αmeff
(p·ej ϕ√3ω3/2
, f) to + αmeff
(p·ej ϕ√3ω3/2
, f).
63
where Rjµ = ejµf/
√ω. Explicitly we can compute as
W ∗+ijR
jµ −W ∗
−ijRjµ = eiν
1√ωTνµ
˜f, W ∗
+ijRjµ −W ∗
−ijRjµ = eiν
1√ωTνµf .
Then
Aµ =1√2
B∗p(e
jν
1√ωTνµf , j) +Bp(e
jν
1√ωTνµ
˜f, j)
+
α
meff
pν(dµνϕ
ω,f
ω)
follows and the theorem is obtained from Lemma 3.22. 2
From Theorem 3.24 we immediately see the corollary below.
Corollary 3.25 (Time evolution of A) Let V = 0. Suppose Assumption 3.4 and
that f is real-valued and f ∈M0 ∩M−1/2. Then
eitHAµ(f)e−itH
=1√2
B∗p(e
itωejν1√ωTνµf , j) +Bp(e
−itωejν1√ωTνµ
˜f, j)
+
α
meff
(−i∇µ)(dµνϕ
ω,f
ω).
Proof: Since eitH =∫ ⊕Rd e
itHpdp,
eitHAµ(f)e−itH =
∫ ⊕RdeitHpAµ(f)e−itHpdp.
Then the corollary follows from Theorem 3.24. 2
3.6 Dressed electron states
Let us study the relationship between the infrared regular/singular condition and
the ground state of Hp.
Definition 3.26 (Dressed electron state) The ground state of Hp is called the
dressed electron state (DES).
Lemma 3.27 Suppose Assumption 3.4. Let Φ be an eigenvector of Hp. Then
Bp(f, j)Φ = 0, j = 1, .., d− 1, for all f ∈M0 ∩M−1/2.
64
Proof: Let HpΦ = EΦ. Then
Bp(f, j)Φ = eitHpe−itHpBp(f, j)eitHpe−itEΦ = eitHpBp(e
itωf, j)e−itEΦ.
Let M = LH∏n
i=1 b∗W (fi, ji)UWΩ|fi ∈ M0 ∩M−1/2, 1 ≤ i ≤ n, n ≥ 0, where UW
denotes the intertwining operator associated with W =
(W+ W−W− W+
). Since a] can
be represented in terms of B]p and
Bp(f, j)UWΩ = −(p · ejQω,f√ω
)UWΩ,
operator a] leaves M invariant, and M is dense in F . Let Ψ =∏n
i=1Bp(fi, ji)UWΩ.
We see that
Bp(eitωf, j)Ψ =
n∑i=1
(e−itωf , fi)B∗p(f1, j1) · · · B∗p(fi, ji) · · ·B∗p(fn, jn)UWΩ
− (p · ejQω, e−itω
f√ω
)Ψ→ 0
as t → ∞ by the Riemann-Lebesgue lemma. By a limiting argument we see that
eitHpBp(eitωf, j)e−itEΦ→ 0 as t→∞. Hence we conclude that Bp(f, j)Φ = 0. 2
Theorem 3.28 (Existence and absence of DES) [Ara83-a, Ara83-b] Suppose
Assumption 3.4.
(1) Let p = 0. Then H0 has a dressed electron state and it is unique.
(2) Suppose in addition that
∫ϕ2
ω3dk < ∞. Then Hp has a dressed electron state
for all p ∈ R, and it is unique.
(3) Suppose
∫ϕ2
ω3dk =∞ and p 6= 0. Then σp(Hp) = ∅. In particular Hp has no
ground state.
Proof: In the case of (1) and (2) we have the unitary equivalence U −1p HpUp =
Hf + Ep. Since the Fock vacuum Ω is the ground state of Hf + Ep, UpΩ is the
65
∫ϕ2
ω3dk <∞
∫ϕ2
ω3dk =∞
p = 0 exist exist
p 6= 0 exist not exist
Figure 3: DES of Hp
dressed electron state of Hp. Next we shall show (3). Let Φ be any bound state of
Hp. Then Bp(f, j)Φ = 0 for all f ∈M0 ∩M−1/2 by Lemma 3.27. Then
0 = (F,Bp(f, j)Φ) = (F, bW (f, j)Φ)− (p · ejQω,f√ω
)(F,Φ)
for F ∈ F such that F ∈ D(N) and (F,Φ) 6= 0. Hence
|(p · ejQω,f√ω
)| ≤ C‖f‖
holds with some constant C, since ‖a](f)F‖ ≤ ‖f‖‖(N + 1l)1/2F‖. Hence the func-
tional f 7→ (p · ej Qω, f√
ω) can be extended on M0(= L2(Rd)). Note that
‖p · ejQ/ω3/2‖2 = p2
∫|ϕ(k)|2
ω(k)3
1
D+(ω(k)2)dk =∞.
The Riesz lemma yields that there exists g ∈ L2(Rd) such that (g, f) = (p ·ej Qω, f√
ω).
It is however contradiction, since p · ej Qω3/2 6∈ L2(Rd). 2
See Figure 3.
3.7 Ground state energy
3.7.1 Holomorphic property
The ground state energy Ep can be represented by W± and ϕ.
66
Lemma 3.29 Suppose Assumption 3.4 and
∫ϕ2
ω3dk <∞. Then8
Ep =1
2m(p+ γ(p))2 + g, (3.88)
where
γ(p)µ = − α2
2meff
pν
(ejνϕ√ω3, (1l +W−W
−1+ )ij
eiµϕ√ω
),
g =1
4m
(ejµϕ√ω, (1l−W−W−1
+ )ijeiµϕ√ω
)Proof: We notice that Ep = (HpΩ,UpΩ)/(Ω,UpΩ),
HpΩ =1
2m
d∑µ=1
((pµ + a∗(fj,µ, j) + a(fj,µ, j))
2Ω,UpΩ),
and Up = exp(a∗(ξj, j) − a(ξj, j))UW , where ξj =1√2
α
meff
p · ej ϕ√ω3
and fj,µ =
− α√2ejµ
ϕ√ω
. Then the lemma follows from Lemma 2.8. 2
We will show that Ep is holomorphic function of α on some neighborhood of
the real line. In what follows in this section we suppose (1),(2) and (3) of Assump-
tion 3.4. Under (1), (2’) and (3) of Assumption 3.4 a similar procedure is also
shown. Set G = ωd−2
2 Gωd−2
2 . Let
H(s) = limε↓0
(d− 1
d
)∫ϕ(k)2
s+ iε− ω(k)2dk. (3.89)
Then D+(s) = m− α2H(s) and
Tµνf = δµνf +
(1
mα2 − H(ω2)
)ϕGϕdµνf. (3.90)
For ζ ∈ C we define Tµν(ζ) by Tµν in (3.90) with m/α2 replaced by ζ, and W±ij(ζ)
by W±ij with Tµν replaced by Tµν(ζ). We see that
8[Hir93, Lemma 5.12] is incorrect. It should be changed to (3.88).
67
Figure 4: ImH(s) and Oε,δ
(1) ImH(s) 6= 0 for s 6= 0,
(2) H(0) = −(d− 1
d
)‖ϕ/ω‖2 < 0,
(3) lims→∞
H(s) = 0.
Let the image of H(s), s ≥ 0, be denoted by H. From (1)-(3) we can see that
for each given ε > 0, there exists δ > 0 such that Oε,δ = x+ iy ∈ C|x > ε, |y| < δsatisfies dist(Oε,δ,H) > 0. See Figure 4. We have
W±ij(ζ) = δij1l + eiµY±µν(ζ)ejν ,
where Y ±µν(ζ) = Y ±(ζ)dµν and
Y +(ζ) =1
ζ − H(ω2)ϕ
(1√ωG√ω +√ωG
1√ω
)ϕT ,
Y −(ζ) =1
ζ − H(ω2)ϕ
(1√ωG√ω −√ωG
1√ω
)ϕ.
68
Let ζ0 ∈ Oε. Then we expand Y ±(ζ) around ζ0 as
Y +(ζ) =∞∑n=0
(−1)n
(ζ0 − H(ω2))n+1ϕ
(1√ωG√ω +√ωG
1√ω
)ϕT
(ζ − ζ0)n,
Y −(ζ) =∞∑n=0
(−1)n
(ζ0 − H(ω2))n+1ϕ
(1√ωG√ω −√ωG
1√ω
)ϕ
(ζ − ζ0)n.
Thus Y ±(ζ) is analytic on Oε,δ, and then so is W±(ζ). Furthermore W±(ζ) ∈ Sp for
ζ ∈ R. Thus we see that
W ∗+(ζ)W+(ζ)−W ∗
−(ζ)W−(ζ) = 1l + ∆1(ζ), (3.91)
W∗+(ζ)W−(ζ)−W ∗
−(ζ)W+(ζ) = ∆2(ζ), (3.92)
where ∆j(ζ), j = 1, 2, are bounded self-adjoint operators with
‖∆j(ζ)‖ < 1 (3.93)
for ζ with sufficiently small imaginary part. Thus for each R > 0 we can define the
open set Oε,δ′,R = x + iy ∈ C|ε < x < R, |y| < δ′ such that Oε,δ′,R ⊂ Oε,δ and
for all ζ ∈ Oε,δ′,R, (3.91), (3.92) and (3.93) hold. Then W−1+ (ζ) exists and is also
holomorphic on Oε,δ′,R. Define
Ep(ζ) =1
2m(p+ γ(p, ζ))2 + g(ζ),
where γ(p, ζ) and g(ζ) are defined with W± replaced by W±(ζ) for ζ ∈ Oε,δ′,R.
Lemma 3.30 Suppose Assumption 3.4 and
∫ϕ2
ω3dk <∞. Then for each ε > 0 and
R > 0, there exits δ′ such that Ep(ζ) is holomorphic on Oε,δ′,R.
Proof: Since
γ(p, ζ)µ = − α2
2meff
pν
(ejνϕ√ω3, (1l +W−(ζ)W+(ζ)−1)ij
eiµϕ√ω
),
g(ζ) =1
4m
(ejµϕ√ω, (1l−W−(ζ)W+(ζ)−1)ij
eiµϕ√ω
),
the theorem follows from the holomorphic properties of W−(ζ) and W+(ζ)−1. 2
69
Lemma 3.31 Suppose Assumption 3.4 and
∫ϕ2
ω3dk <∞. Then there exists m∗ > 0
such that Ep =1
2m∗p2 + g.
Proof: Notice that
γ(p)µ = − α2
2meff
pν(ejνϕ√ω3, (1l +W−W
−1+ )ij
eiµϕ√ω
)
= − α2
meff
pµ
(d− 1
d
)‖ϕ/ω‖2 − α2
2meff
pν(ejνϕ√ω3, eiaY
−abe
kb (W
−1+ )kj
eiµϕ√ω
).
Let α2 be sufficiently small. We notice that W+ = 1l + Y, where Yij = eiµY+µνe
jν . The
second term is computed as
(ejνϕ√ω3, eiaY
−abe
kb (W
−1+ )kj
eiµϕ√ω
) = (ejνϕ√ω3, eiaY
−abe
kb ((1l + Y)−1)kj
eiµϕ√ω
)
=∞∑n=0
(−1)n(ejνϕ√ω3, eiaY
−abe
kb (Y
n)kjeiµϕ√ω
).
Thus we directly see that
(ejνϕ√ω3, eiaY
−abe
kb (W
−1+ )kj
eiµϕ√ω
)
=∞∑n=0
(−1)n(eiνϕ√ω3, eiaY
−abe
kbekµ1Y +µ1ν1
ek1ν1ek1µ2Y +µ2ν2
ek2ν2· · · ekn−1
µn Y +µnνne
knνn
eiµϕ√ω
)
=∞∑n=0
(−1)n(ϕ√ω3, daνY
−dabdbµ1Y+dµ1ν1dν1µ2Y
+dµ2ν2dν2µ3Y+ · · ·Y +dµnνndνnµ
ϕ√ω
)
= δµν
∞∑n=0
(−1)n(d− 1
d
)n+2
(ϕ√ω3, Y −Y +n ϕ√
ω).
We set the right hand side by δµνM . Then we have
Ep =1
2m(p+ γ(p))2 + g =
p2
2m
(1− α2
meff
(d− 1
d
)‖ϕ/ω‖2 − α2
2meff
M
)2
+ g.
Hence the corollary follows for sufficiently small α2. Since Ep is holomorphic on
Oε,δ,R for arbitrary ε > 0 and R > 0, the corollary follows for all α ∈ R. 2
70
In Lemma 3.31 we suppose
∫ϕ2
ω3dk < ∞. This condition is, however, removed
in the next section.
3.7.2 Explicit form of effective mass and ground state energy
In the present section we show that Ep is of the form 12m∗
p2 + g. The main theorem
in this section is as follows.
Theorem 3.32 (Explicit form of Ep) Suppose Assumption 3.4. Then Ep =1
2meff
p2 + g for all α ∈ R, where g is given by
g =d
2π
∫ ∞−∞
α2(d−1d
) ∥∥ tϕt2+ω2
∥∥2
m+ α2(d−1d
) ∥∥∥ ϕ√t2+ω2
∥∥∥2dt.
Note that we do not assume
∫ϕ2
ω3dk <∞ in Theorem 3.32. Throughout this section
we assume that α2 is sufficiently small unless otherwise stated.
Since a momentum lattice approximated Hp can be identified with a harmonic
oscillator in L2(RD) for some D, Ep can be obtained through calculating the ground
state energy of the harmonic oscillator.
First ω is replaced by ωε(k) = ω(k) + ε for ε > 0. For l = (l1, · · · , ld) ∈Rd, let |l| = maxj |lj|. For the time being we suppose l ∈ (2πZ/a)d, |l| ≤ 2πL
with some a and L; l is a lattice point with the width 2π/a of the d-dimensional
rectangle centered at the origin with the width 4πL. The lattice points are named
l1, l2, · · · , l`, where ` = (2[aL] + 1)d denotes the number of lattice points and [z]
denotes the integer part of z ∈ R. For l ∈ (2πZ/a)d, we define the rectangle:
Γ(l) =[l1, l1 + 2π
a
)× · · · ×
[ld, ld + 2π
a
). Let
Qjl =1√2
1√ωε(l)
( a2π
)d/2 a∗(χΓ(l), j) + a(χΓ(l), j)
, (3.94)
Pjl =i√2
√ωε(l)
( a2π
)d/2 a∗(χΓ(l), j)− a(χΓ(l), j)
. (3.95)
Then the Weyl relations hold,
exp (itPjl) exp (isQj′l′) = exp(itsδl1l′1 ...δldl′dδjj′
)exp (isQj′l′) exp (itPjl) , t, s ∈ R.
(3.96)
71
Let D = (d− 1)`. We define the D ×D-diagonal matrix by9
A0 =
ωε(l1)21l
ωε(l2)21l. . .
ωε(l`)21l
,
where 1l denotes the (d− 1)× (d− 1)-identity matrix. Since ε > 0, A0 is a strictly
positive matrix. We denote by (f, g)D the D-dimensional scalar product. Let vµjl =
ϕ(l)ejµ(l), and
~vµ =
vµ1l1...
vµd−1l1...vµ1l`
...vµd−1l`
∈ RD, µ = 1, ..., d.
For linear operator T , let 〈T 〉D = (~vµ, T~vµ)D. Suppose that T : Rd → R is a rotation
invariant function. Let Tdiag be the D ×D-diagonal matrix with diagonal elements
T (l):
Tdiag =
T (l1)1l
T (l2)1l. . .
T (l`)1l
.
Then (~vµ, Tdiag~vν)D = δµν(d−1d
)∑|l|≤2πL T (l)|ϕ(l)|2. Let P = (Pjl)1≤j≤d−1,|l|≤2πL and
Q = (Qjl)1≤j≤d−1,|l|≤2πL. Then the momentum lattice approximated Hp is written
as
HεL,a(p) =
1
2m(p− α(~v,Q)D)2 +
1
2((P, P )D + (Q,A0Q)D)− tr
√A0,
where p ∈ Rd and (~v,Q)D = ((~v1, Q)D, ..., (~vd, Q)D).
9A0 given in [HS01, p.1176 in Appendix] is incorrect. Matrix elements ωε(lj) are changed toωε(lj)
2.
72
Lemma 3.33 Suppose that ε > 0. Let ~p and ~q be the momentum operator and its
canonical position operator in L2(RD), respectively. Then there exist10 M ≤ ∞, a
D ×D nonnegative symmetric matrix A and ~f ∈ RD such that
HεL,a(p)
∼=M⊕(
1
2(~p, ~p)D +
1
2(~q, A~q)D +
1
2mp2 − 1
2(~f,A~f)D −
1
2tr√A0
). (3.97)
Proof: Define the D × D-matrix by P =∑d
µ=1 |~vµ〉〈~vµ|. Set A = A0 + α2
mP . Note
that A is a strictly positive symmetric matrix, since A0 is strictly positive and P is
nonnegative. In particular, (A+ a)−1 exists for a ≥ 0. Let ~f = ~f(p) = αmA−1pµ~vµ ∈
RD. Then we have
HεL,a(p) =
1
2(P, P )D +
1
2((Q− ~f), A(Q− ~f))D +
1
2mp2 − 1
2(~f,A~f)D −
1
2tr√A0.
By (3.96) and the von Neumann uniqueness theorem, there exists M ≤ ∞ and a
unitary operator U : F →⊕M L2(RD) implementing that
UPjlU−1 =
M⊕(−i∇xjl),
UQjlU−1 =
M⊕xjl.
Then HεL,a(p) is unitarily equivalent with the direct sum of the harmonic oscillator:
M⊕(1
2(~p, ~p)D +
1
2((~q − ~f), A(~q − ~f))D +
1
2mp2 − 1
2(~f,A~f)D −
1
2tr√A0
)in⊕M L2(RD). By the shift ~q → ~q + ~f implemented by a unitary operator, we
obtain (3.97). 2
Lemma 3.34 Suppose ε > 0. Then
infσ(HεL,a(p)) =
1
2mp2 − 1
2(~f,A~f)D +
1
2tr(√A−
√A0). (3.98)
10Possibly M =∞.
73
Proof: Generally for the harmonic oscillator HT = 12(~p, ~p)D + 1
2(~q, T~q)D with a sym-
metric nonnegative matrix T , infσ(HT ) = 12tr√T . Hence
infσ
(1
2(~p, ~p)D +
1
2(~q, A~q)D
)=
1
2tr√A.
Thus the ground state energy of HεL,a(p) is given by (3.98) by Lemma 3.33. 2
We calculate (~f,A~f)D and tr(√A−√A0) as follows. We note that
(~vµ, A−10 ~vν) = δµν
(d− 1
d
) ∑|l|≤2πL
|ϕ(l)|2
ωε(l)2, (3.99)
(~vµ, (s2 + A0)−1~vν) = δµν
(d− 1
d
) ∑|l|≤2πL
|ϕ(l)|2
s2 + ωε(l)2, (3.100)
(~vµ, (s2 + A0)−1A0~vν) = δµν
(d− 1
d
) ∑|l|≤2πL
ωε(l)2|ϕ(l)|2
s2 + ωε(l)2. (3.101)
Furthermore A−1 = s− limN→∞
N∑n=1
(−α2
mA−1
0 P )n−1A−10 .
Lemma 3.35 Suppose ε > 0. Then
1
2mp2 − 1
2(~f,A~f)D =
p2
2m
1
1 + α2
mθ, (3.102)
where θ =
(d− 1
d
) ∑|l|≤2πL
|ϕ(l)|2
ωε(l)2.
74
Proof: By (3.99) we have
(~f,A~f)D
=1
m
α2
mpµpν(~vµ, A
−1~vν)D
=1
m
α2
mpµpν
∞∑n=1
(−α2
m)n−1(~vµ, (A
−10 P )n−1A−1
0 ~vν)D
=1
m
α2
m
∞∑n=1
pµpν(−α2
m)n−1(~vµ, A
−10 ~vµ1)D(~vµ1 , A
−10 ~vµ2)D · · · (~vµn−1 , A
−10 ~vν)D
=1
m
α2
m
∞∑n=1
pµpνδµµ1δµ1µ2 · · · δµn−1ν(−α2
m)n−1θn
=1
m
α2
m
∞∑n=1
(−α
2
m
)n−1
p2θn
=α2
mθ
1 + α2
mθ
p2
m.
Hence (3.102) follows. 2
Lemma 3.36 Suppose ε > 0. Then
1
2tr(√
A−√A0
)=d− 1
2π
∫ ∞−∞
α2
ms2
1 + α2
mξ
∑|l|≤2πL
|ϕ(l)|2
(s2 + ωε(l)2)2ds, (3.103)
where ξ =
(d− 1
d
) ∑|l|≤2πL
|ϕ(l)|2
s2 + ωε(l)2.
Proof: We see that
tr√A− tr
√A0 =
1
π
∫ ∞−∞
trA(s2 + A)−1 − A0(s2 + A0)−1
ds.
Let A∞ =∑∞
n=1
−α2
mP (s2 + A0)−1
n. We have
A(s2 + A)−1 − A0(s2 + A0)−1 =α2
mP (s2 + A0)−1 + A(s2 + A0)−1A∞.
75
It follows that
trα2
mP (s2 + A0)−1 =
α2
m
d∑µ=1
∑φ:CONS
(φ,~vµ)D(~vµ, (s2 + A0)−1φ)D,
where∑
φ:CONS means to sum up all the vectors φn in a complete orthonormal
system (CONS). Take a CONS such that
φ1 =
~vµ‖~vµ‖
, φ2, φ3, · · ·
. Then we have
by (3.100)
trα2
mP (s2 + A0)−1 =
α2
mε(s2 + A0)−1 = d
α2
mξ. (3.104)
We see that
A(s2 + A0)−1A∞ = A0(s2 + A0)−1A∞ +α2
mP (s2 + A0)−1A∞.
It follows that
trA0(s2 + A0)−1A∞
=∞∑n=1
(−α2
m)n
∑φ:CONS
((s2 + A0)−1A0φ, (P (s2 + A0)−1)nφ
)D
=∞∑n=1
(−α2
m)n
∑φ:CONS
((s2 + A0)−1A0φ,~vµ1)D×
× ((s2 + A0)−1~vµ1 , ~vµ2)D · · · ((s2 + A0)−1~vµn , φ)D.
76
Take a CONS such that
φ1 =
(s2 + A0)−1A0~vµn‖(s2 + A0)−1A0~vµn‖
, φ2, φ3, · · · ,
. Since |α| is suf-
ficiently small, from (3.101) it follows that
trA0(s2 + A0)−1A∞ =∞∑n=1
(−α
2
m
)n((s2 + A0)−1~vµn , (s
2 + A0)−1A0~vµ1)D×
× ((s2 + A0)−1~vµ1 , ~vµ2)D · · · ((s2 + A0)−1~vµn−1 , ~vµn)D
=∞∑n=1
(−α
2
m
)nδµ1µ2 · · · δµn−1µnξ
n−1ε(s2 + A0)−2A0
= −α2
m
ε(s2 + A0)−2A0
1 + α2
mξ
=−dα2
m
1 + α2
mξ
∑|l|≤2πL
ωε(l)2|ϕ(l)|2
(s2 + ωε(l)2)2, (3.105)
and
trα2
mP (s2 + A0)−1A∞
=∞∑n=1
(−α2
m)nα2
m
∑φ:CONS
(φ, P (s2 + A0)−1
(P (s2 + A0)−1
)nφ)D
=∞∑n=1
(−α2
m)nα2
m
∑φ:CONS
(φ,~vµ1)D(~vµ1 , (s2 + A0)−1~vµ2)D · · · (~vµn+1 , (s
2 + A0)−1φ)D.
Take a CONS such thatφ1 =
~vµ1
‖~vµ1‖, φ2, φ3, · · · ,
. Then we see that
trα2
mP (s2 + A0)−1A∞ = −
∞∑n=1
(−α2
m)n+1δµ1µ2 · · · δµnµn+1δµn+1µ1ξ
n+1 = d−α2
m
2ξ2
1 + α2
mξ.
(3.106)
77
Hence we have
trA(s2 + A)−1 − A0(s2 + A0)−1
=ε(s2 + A0)−2A0
1 + α2
mξ
(−α
2
m
)− d
(α2
mξ)2
1 + α2
mξ
+ dα2
mξ
=α2
m
d(d−1d
)1 + α2
mξ
∑|l|≤2πL
|ϕ(l)|2
s2 + ωε(l)2− ωε(l)
2|ϕ(l)|2
(s2 + ωε(l)2)2
=(d− 1)α
2
ms2
1 + α2
mξ
∑|l|≤2πL
|ϕ(l)|2
(s2 + ωε(l)2)2.
Thus the lemma follows. 2
Lemma 3.37 Suppose the same assumptions as in Lemma 3.33. Then
infσ(HεL,a(p)) =
p2
2(m+ α2θ)+d− 1
2π
∫ ∞−∞
α2s2
m+ α2ξ
∑|l|≤2πL
|ϕ(l)|2
(s2 + ωε(l)2)2ds.
Proof: It follows from Lemmas 3.35 and 3.36. 2
The proof of Theorem 3.32
Proof: Suppose that
∫ϕ2
ω3dk <∞ and α2 is sufficiently small. We set
meff(a, L, ε) = m+ α2θ,
g(a, L, ε) =d− 1
2π
∫ ∞−∞
α2s2
m+ α2ξ
∑|l|≤2πL
|ϕ(l)|2
(s2 + ωε(l)2)2ds.
Note that meff(a, L, ε)→ meff and g(a, L, ε)→ g as a→∞, L→∞, ε→ 0. Taking
a→∞ and then L→∞, we see that HεL,a(p)→ Hp+ εN uniformly in the resolvent
sense, which yields that infσ(HεL,a(p))→ infσ(Hp + εN). Hence
infσ(Hp + εN) = limL→∞
lima→∞
(p2
2meff(a, L, ε)+ g(a, L, ε)
)=
p2
2meff(ε)+ g(ε),
78
where meff(ε) and g(ε) are defined by meff and g with ω replaced by ωε. Since
Hp + εN → Hp strongly on D(Hf) as ε → 0, Hp + εN → Hp holds in the strong
resolvent sense. Then it follows
lim supε→0
infσ(Hp + εN) ≤ infσ(Hp). (3.107)
Furthermore, since N ≥ 0, we have
lim infε→0
infσ(Hp + εN) ≥ infσ(Hp). (3.108)
Combining (3.107) and (3.108) we have infσ(Hp + εN)→ infσ(Hp) = Ep as ε→ 0.
Then
Ep = limε→0
(p2
2meff(ε)+ g(ε)
)=
p2
2meff
+ g.
Hence the theorem follows for sufficiently small α2. The theorem is valid however for
all α since Ep is holomorphic on Oε,δ′,R for arbitrary ε > 0 and R > 0 by Lemma 3.31.
Next we do not assume
∫ϕ2
ω3dk <∞. Let ϕn be a sequence such that ϕnω
l → ϕ/ωl
for l = 0,−1/2,−1 and ϕn/ω3/2 ∈ L2(Rd). Then the ground state energy of Hp with
cutoff function ϕn, Hp(n), is given by
1
2(m+ α2(d−1d
)‖ϕn/ω‖2)
p2 +d
2π
∫ ∞−∞
α2(d−1d
) ∥∥ tϕnt2+ω2
∥∥2
m+ α2(d−1d
) ∥∥∥ ϕn√t2+ω2
∥∥∥2dt. (3.109)
We can see that Hp(n) → Hp as n → ∞ in the uniform resolvent sense. Then
infσ(Hp(n))→ infσ(Hp) and (3.109) converges to Ep as n→∞. Hence the theorem
follows.
3.7.3 Ultraviolet cutoffs
In this subsection we assume that d = 3, α = 1, the ultraviolet cutoff is given by
the sharp cutoff
ϕ(k) = (2π)−3/21l[λ,Λ](k). (3.110)
See Example 3.10. Set g = g(Λ) for the emphasis of the dependence of the ultraviolet
cutoff parameter Λ. We estimate the asymptotic behavior of g(Λ) as Λ → ∞. In
79
the case of V = 0, from Theorem 3.6 it follows that g(Λ) = infσ(H). It is seen that
g(Λ) is monotonously increasing in Λ. Since
‖ϕ/√t2 + ω2‖2 = 4π
(Λ− λ) + t
(arctan
λ
t− arctan
Λ
t
),
and
‖ϕ/(t2 + ω2)‖2 = 4π
1
2t
(arctan
Λ
t− arctan
λ
t
)+
1
2
(λ
t2 + λ2− Λ
t2 + Λ2
),
changing variable t to r = Λ/t, we have the explicit form of ground state energy of
H with ultraviolet cutoff (3.110) and V = 0.
Proposition 3.38 (Ground state energy) Suppose that V = 0, d = 3, (3.110),
and α = 1. Then
g(Λ) = 4Λ2
∫ ∞0
(arctan r − r1+r2 )−
(arctan r
(λΛ
)− r( λΛ)
1+r2( λΛ)2
)mr + 8π
3Λ
(r − arctan r)− (r(λΛ
)− arctan r
(λΛ
)) drr2. (3.111)
Theorem 3.39 (Asymptotic behavior of g(Λ)) Assume that m > 8πλ/3. Then
8
3
(3
8π
1
m
)1/2π
2≤ lim
Λ→∞
g(Λ)
Λ3/2≤ 8
3
(9
8π
1
m
)1/2π
2.
Proof: We decompose g(Λ)4Λ
as g(Λ)4Λ
=∫ 1/Λ1/4
0+∫∞
1/Λ1/4 = I1(Λ) + I2(Λ). It is enough
to show that2
3
(3
8π
1
m
)1/2π
2≤ lim
Λ→∞
I1(Λ)√Λ≤ 2
3
(9
8π
1
m
)1/2π
2, (3.112)
and that limΛ→∞I2(Λ)√
Λ= 0. Note that arctanx = x
1+x2 + 23
x3
(1+x2)2 + 54
32
x5
(1+x2)3 + · · · .We define functions f and h by
arctan r − r
1 + r2=
2
3
r3
(1 + r2)2+ f(r),
r − arctan r =r3
1 + r2− h(r).
80
It is satisfied that f(r) ≥ 0, h(r) ≥ 0, limr→0f(r)r3 = 0, limr→0
h(r)r3 = 2
3and
h(r) =2
3
r3
(1 + r2)2+ f(r) = arctan r − r
1 + r2.
Let us set
fΛ(r) = arctan r
(λ
Λ
)−
r(λΛ
)1 + r2
(λΛ
)2 > 0,
hΛ(r) = r
(λ
Λ
)− arctan r
(λ
Λ
)> 0.
Then I1(Λ) is written as
I1(Λ) =
∫ 1/Λ1/4
0
23
+ (1+r2)2
r3 (f(r)− fΛ(r))mΛ
(1 + r2) + 8π3r2 − 8π
31+r2
r(h(r) + hΛ(r))
dr
1 + r2.
It follows that for 0 ≤ r ≤ 1/Λ1/4,
8π
3
1 + r2
r(h(r) + hΛ(r)) =
8π
3r2(1 + r2)
h(r) + hΛ(r)
r3≤ 8π
3
(1 +
1√Λ
)r2θ(Λ),
(3.113)
where θ(Λ) = sup0≤r≤1/Λ1/4h(r)+hΛ(r)
r3 . We set the right hand side of (3.113) by r2δ(Λ).
Note that
limr↓0
h(r) + hΛ(r)
r3=
2
3+
1
3
(λ
Λ
)3
, limΛ→∞
δ(Λ) =8π
3
2
3. (3.114)
Moreover we have
− sup0≤r≤(1/Λ1/4)
(1 + r2)2
r3fΛ(r) ≤ (1 + r2)2
r3(f(r)− fΛ(r)) ≤ sup
0≤r≤(1/Λ1/4)
(1 + r2)2
r3f(r).
Set ε(Λ) = max
sup0≤r≤(1/Λ1/4)(1+r2)2
r3 f(r), sup0≤r≤(1/Λ1/4)(1+r2)2
r3 fΛ(r)
. It is trivial
to see that limΛ→∞ ε(Λ) = 0. Hence we have
23− ε(Λ)
1 + 1/√
Λ
∫ 1/Λ1/4
0
drmΛ
+(mΛ
+ 8π3
)r2≤ I1(Λ) ≤
23
+ ε(Λ)
1− 1/√
Λ
∫ 1/Λ1/4
0
drmΛ
+(mΛ
+ 8π3− δ(Λ)
)r2.
81
Then a direct calculation yields that
limΛ→∞
1√Λ
∫ 1/Λ1/4
0
1mΛ
+(mΛ
+ 8π3− δ(Λ)
)r2dr
= limΛ→∞
1√m(m
Λ+ 8π
3− δ(Λ))
arctan
√mΛ
+ 8π3− δ(Λ)
m/√
Λ
=
(9
8π
1
m
)1/2π
2.
Similarly we have
limΛ→∞
1√Λ
∫ 1/Λ1/4
0
1mΛ
+(mΛ
+ 8π3
)r2dr =
(3
8π
1
m
)1/2π
2.
Thus2
3
(3
8π
1
m
)1/2π
2≤ lim
Λ→∞
1√ΛI1(Λ) ≤ 2
3
(9
8π
1
m
)1/2π
2.
Hence (3.112) follows. Next we show that limΛ→∞
I2(Λ)√Λ
= 0. Since
(arctan r − r
1 + r2)−
(arctan r
(λ
Λ
)−
r(λΛ
)1 + r2
(λΛ
)2
)≤ 2
3
r3
(1 + r2)2+
5
4
3
2
r5
(1 + r2)2
and by the assumption m > 8πλ/3,
m
Λr +
8π
3
(r − arctan r)−
(r
(λ
Λ
)− arctan r
(λ
Λ
))>m
Λr − λ
Λ
8π
3r +
8π
3
r3
1 + r2− 2
3
r3
1 + r2+ arctan(r
(λ
Λ
))
>
8π
9
r3
1 + r2.
Then
limΛ→∞
1√ΛI2(Λ) ≤ lim
Λ→∞
1√Λ
∫ ∞1/Λ1/4
23
r3
(1+r2)2 + 54
32
r5
(1+r2)2
8π9
r3
1+r2
dr
r2
= limΛ→∞
9
8π
1√Λ
∫ ∞1/Λ1/4
1
1 + r2
(2
3+
15
8r2
)dr
r2= 0.
Thus limΛ→∞I2(Λ)√
Λ= 0 follows, and then the theorem is proven. 2
82
3.7.4 Many particle system
We next consider an N particle system. We assume simply that each particle has
mass m and there is no external potential. The Hamiltonian, H, is defined as a
self-adjoint operator acting on L2(R3N)⊗F , and is given by
HN =N∑j=1
1
2m(−i∇j − αAj)2 +Hf , (3.115)
where
Aj,µ =1√2
∫ej′µ (k)√ω(k)
ϕj(k)a∗(k, j′) + ϕj(k)a(k, j′) dk.
Let infσ(HN) = g(Λ, N). We consider the two cases:
(1) ϕj = ϕ, j = 1, ..., N,
(2) suppϕj ∩ suppϕi ∩ 0 = ∅, i 6= j.
We will see below that the asymptotic behavior of g(Λ, N) as N → ∞ depends on
ultraviolet cutoffs. In the case of (2) we intuitively expect that g(Λ, N) ≈ N , since
N particles may have no interaction through quantized radiation fields.
Proposition 3.40 (Ground state energy for many particle system) In the
case of (1),
g(Λ, N) =N
π
∫ ∞−∞
‖tϕ/(t2 + ω2)‖2
m+ 23N‖ϕ/
√t2 + ω2‖2
dt,
in the case of (2),
g(Λ, N) =N∑j=1
1
π
∫ ∞−∞
‖tϕj/(t2 + ω2)‖2
m+ 23‖ϕj/√t2 + ω2‖2
dt.
Proof: This is proven in a similar manner to Theorem 3.32. 2
In the case of (1) the following theorem holds.
Theorem 3.41 (Asymptotic behavior of g(Λ)) We assume case (1) and m >
8πλ/3. Then
8
3
(3
8π
1
m
)1/2π
2≤ lim
Λ,N→∞
g(Λ, N)√NΛ3/2
≤ 8
3
(9
8π
1
m
)1/2π
2.
83
Proof: By Proposition 3.40 we have
g(Λ, N)
4Λ=
∫ ∞0
(arctan r − r1+r2 )−
(arctan r
(λΛ
)− r( λΛ)
1+r2( λΛ)2
)mNΛr + 8π
3
(r − arctan r)− (r
(λΛ
)− arctan r
(λΛ
)) drr2.
Then in the similar way as the proof of Theorem 3.41 we decompose it such as
g(Λ, N)
4Λ=
∫ 1/(ΛN)1/4
0
+
∫ ∞1/(ΛN)1/4
= I1(Λ, N) + I2(Λ, N),
and it can be seen that
2
3
(3
8π
1
m
)1/2π
2≤ lim
Λ,N→∞
I1(Λ, N)√NΛ
≤ 2
3
(9
8π
1
m
)1/2π
2,
and that limΛ,N→∞I2(Λ,N)√
NΛ= 0. Then the theorem follows. 2
3.8 Self-energy term
3.8.1 Diagonalization and DES
In this section we investigate the A2-dependence of the ground state energy and
DES. Let us define
Hε = Hp −α
m(−i∇) · A+ ε
α2
2mA2 +Hf + V, (3.116)
where 0 ≤ ε ≤ 1 denotes a parameter. In the case of ε = 1, Hε describes H
and in the case of ε = 0, H without self-energy term A2. So the parameter ε
interpolates between them. Neglecting external potential V , we first study the
translation invariant Hamiltonian defined by
Hε,p =1
2mp2 − α p
m· A+ ε
α2
2mA2 +Hf , p ∈ Rd. (3.117)
We can also diagonalize Hε,p in a similar manner to Hp. Let
Dε(z) = m− εα2
2
(d− 1
d
)∫ϕ(k)2
z − ω(k)2dk. (3.118)
84
Then we define Dε+(s) and Qε by D+(s) and Q, respectively, with D(z) replaced
by Dε(z). We also define T εµνf = δµνf + εαQεωd−2
2 Gωd−2
2 dµνϕf . Furthermore the
ground state energy Eε,p of Hε,p can be explicitly computed in the same manner as
that of H. The net result is as follows:
Eε,p =1
2mε
p2 + gε, (3.119)
where
1
mε
=1
m−
α2(d−1d
)‖ϕ/ω‖2
m+ εα2(d−1d
)‖ϕ/ω‖2
1
m,
gε =d
2π
∫ ∞−∞
εα2(d−1d
) ∥∥ tϕt2+ω2
∥∥2
m+ εα2(d−1d
) ∥∥∥ ϕ√t2+ω2
∥∥∥2dt.
By replacing Tµν with T εµν , we define W ε =
(W ε
+ W ε−
W ε− W ε
+
)∈ Sp2 and the intertwining
operator U (W ε). On the other hand the displacement operator is given by e−iΠε
under the assumption
∫ϕ2
ω3dk <∞, where
Πε =i√2
α
mεeff
a∗(
p · ejϕω3/2
, j)− a(p · ejϕω3/2
, j)
, p ∈ Rd, (3.120)
with mεeff = m + εα2
(d−1d
)‖ϕ/ω‖2. Then we define Uε,p by Uε,p = e−iΠεUW ε for
p ∈ Rd.
Theorem 3.42 (Diagonalization of Hε,p) Suppose Assumption 3.4.
(1) Let p = 0. Then
U −1ε,0
(εα2
2mA2 +Hf
)Uε,0 = gε +Hf . (3.121)
(2) Suppose
∫ϕ2
ω3dk <∞. Then
U −1ε,p Hε,pUε,p =
1
2mε
p2 + gε +Hf . (3.122)
85
Proof: The proof is similar to Theorem 3.5. 2
We define the unitary operator Uε on H by
Uε =
∫ ⊕Rd
Uε,peiπ
2Ndp. (3.123)
Theorem 3.43 (Diagonalization of Hε) Suppose Assumption 3.4 and that V is
infinitesimally small with respect to −∆. Assume furthermore that
∫ϕ2
ω3dk < ∞.
Then Hε is self-adjoint on D(−∆)∩D(Hf), and for each α ∈ R, Uε maps D(−∆)∩D(Hf) onto itself and
Uε−1HUε = − 1
2mεeff
∆ +Hf + T−1ε V Tε + gε, (3.124)
where
Tε = exp (−i(−i∇) ·Kε) , (3.125)
Kεµ =
1√2
∫ejµ(k)√ω(k)
(Qε(k)
ω(k)a∗(k, j) +
Qε(k)
ω(k)a(k, j)
)dk. (3.126)
Proof: We set Hε with V = 0 by Hε(0). In a similar way to Proposition 3.2 we can
show that Hε(0) is self-adjoint on D(−∆) ∩ D(Hf). By the closed graph theorem
there exists C > 0 such that ‖(−∆ + Hf)F‖ ≤ C(‖Hε(0)Φ‖ + ‖Φ‖). Hence V is
infinitesimally small with respect to Hε(0) and Hε is self-adjoint on D(−∆)∩D(Hf)
by the Kato-Rellich theorem. Statement (3.124) is proven in a similar manner to
Theorem 3.6. 2
Corollary 3.44 (Existence and absence of DES of Hε,p)
Suppose Assumption 3.4. Then (1)-(3) follow:
(1) The Hamiltonian Hε,p for p = 0 has a dressed electron state and it is unique.
(2) Suppose
∫ϕ2
ω3dk < ∞. Then Hε,p has a dressed electron state for all p ∈ Rd
and it is unique.
(3) Suppose
∫ϕ2
ω3dk =∞. Then Hε,p with p 6= 0 has no bound state.
86
Proof: The proof is similar to that of Theorem 3.6. 2
Corollary 3.45 Let 0 ≤ ε < 1. Suppose Assumption 3.4 and that V is nonnegative
and infinitesimally small with respect to −∆. Assume also that
∫ϕ2
ω3dk <∞. Let
α∗ =m
1− ε1(
d−1d
)‖ϕ/ω‖2
.
Then Hε is bounded from below for α2 < α∗ and unbounded from below for α2 > α∗.
In particular Hε has no ground state for α2 > α∗.
Proof: Let Hε(0) be Hε with V = 0. When α2 < α∗ (resp. α2 > α∗),1mε
> 0
(resp. 1mε
< 0) follows. Then Hε(0) is bounded from below (resp. unbounded from
below). Since V ≤ 0, Hε is also unbounded from below for α2 > α∗. Contrary to
this, Hε is bounded from below for α2 < α∗, since T−1V T is infinitesimally small
with respect to − 12mε
∆ +Hf . 2
3.8.2 No self-enery term
We consider the special case: ε = 0. Thus Hamiltonians are
H0 = − 1
2m∆− α
m(−i∇) · A+Hf + V, (3.127)
H0,p =1
2mp2 − α
mp · A+Hf . (3.128)
Then Dε(z) = m, Qε = αϕ/m, T εµν = δµν , g = 0, W ε+ = 1l and W ε
− = 0. Thus the
intertwining operator is the identity and U0,p = e−iΠ, where
Π =i√2
α
m
a∗(
p · ejϕω3/2
, j)− a(p · ejϕω3/2
, j)
. (3.129)
Suppose
∫ϕ2
ω3dk <∞. Then
U −10,p H0,pU0,p =
1
2m0
p2 +Hf . (3.130)
87
Let U0 =∫
U0,pei pi
2Ndp. Then
U −10 (− 1
2m∆− α
m(−i∇) · A+Hf + V )U0 = − 1
2m0
∆ +Hf + T−1V T, (3.131)
where1
m0
=1
m−α2(d−1d
)‖ϕ/ω‖2
m(3.132)
and T = exp(−i(−i∇) ·K) with
Kµ =1√2
α
m
∫ejµ(k)√ω(k)3
(ϕ(k)a∗(k, j) + ϕ(k)a(k, j)) dk.
Thus formally we have T−1V T = V (x+ αmZ), where
Zµ =1√2
∫ejµ(k)√ω(k)3
(ϕ(k)a∗(k, j) + ϕ(k)a(k, j)) dk.
Let V be infinitesimally small with respect to −∆. By (3.132) we see that H0
is unbounded from below for α2 > ((d−1d
)‖ϕ/ω‖2)−1 and bounded from below for
α2 < ((d−1d
)‖ϕ/ω‖2)−1.
3.9 Scaling limits
In this section we investigate scaling limits of Hamiltonian H and derive effective
Hamiltonians. The general references in this section are [Ara90, Dav77, Dav79,
Hir93, Hir97, Hir98, Hir99, Hir02].
3.9.1 Weak coupling limit
It is shown that one of the useful tool to derive effective objects is scaling limits.
We introduce the scaling by a] → κa]. Then the scaled Hamiltonian is of the form
H(κ) =1
2m(−i∇− ακA)2 + V + κ2Hf . (3.133)
We consider the asymptotic behavior of H(κ) as κ → ∞. The scaling (3.133) is
equivalent to the substitution
ω → κ2ω, ϕ→ κ2ϕ. (3.134)
88
The operator Tµν leaves invariant under this scaling, and hence W± and the inter-
twining operator UW also leave invariant under (3.134). On the other hand the
displacement operator Sp is scaled as
Sp → exp
(1
κ
1√2
α
meff
a∗(
p · ejϕω3/2
, j)− a(p · ejϕω3/2
, j)
)(3.135)
and then we have
s-limκ→∞
Sp = 1l. (3.136)
Let U =∫ ⊕
SpUW eiπ
2Ndp. We recall that g = d
2π
∫∞−∞
α2( d−1d )
∥∥∥ tϕ
t2+ω2
∥∥∥2
m+α2( d−1d )
∥∥∥∥ ϕ√t2+ω2
∥∥∥∥2dt. Then
g is scaled as g → κ2g. If we make the corresponding substitution in U and δV , we
denote them by Uκ and δVκ, respectively.
Lemma 3.46 It follows that s-limκ→∞
Uκ = UW .
Proof: The lemma follows from (3.136) and the invariance of UW under (3.134). 2
In order to consider the asymptotic behavior of H(κ) as κ → ∞, we introduce
the energy renormalization κ2g.
Theorem 3.47 (Weak coupling limit) [Hir02] Let V be relatively bounded with
respect to − 12m
∆ with a relative bound strictly smaller than one. Suppose Assump-
tion 3.4 and
∫ϕ2
ω3dk <∞. Then for z ∈ C \ R,
s-limκ→∞
(H(κ)− κ2g − z)−1 = (Heff − z)−1 ⊗ PW , (3.137)
where PW denotes the projection to the one dimensional subspace aUWΩ|a ∈ C.
Proof: By the unitary transformation Uκ we have
(H(κ)− κ2g − z)−1 = Uκ(Heff + κ2Hf + δVκ − z)−1U −1κ . (3.138)
We have already seen that s-limκ→∞Uκ = UW . We can directly see that
(1) D(δVκ) ⊃ D(Heff) and δVκ(Heff + λ)−1 is bounded in H for large λ > 0 with
limλ→∞ ‖δVκ(Heff + λ)−1‖ = 0,
89
(2) δVκ(Heff + λ)−1 is strongly continuous in κ,
(3) s-limκ→∞ δVκ(Heff + λ)−1 = 0.
In the abstract formula in [Ara90], it has been established that (1)-(3) above imply
that
s-limκ→∞
(Heff + κ2Hf + δVκ − z)−1 = (Heff − z)−1 ⊗ PΩ, (3.139)
where PΩ denotes the projection to aΩ|a ∈ C. Hence we can see that
s-limκ→∞
(H(κ)− κ2g − z)−1 = s-limκ→∞
Uκ(Heff + κ2Hf + δVκ − z)−1U −1κ
= UW
((Heff − z)−1 ⊗ PΩ
)U −1W
= (Heff − z)−1 ⊗ (UWPΩU −1W )
= (Heff − z)−1 ⊗ PW .
Then the theorem is proven. 2
We notice that the projection PW denotes the projection to the one dimensional
subspace spanned by the unique ground state of operator Hp=0 = 12mA2 +Hf .
3.9.2 Strong coupling limit
In the previous section we study the weak coupling limit which is given by the
asymptotic behavior of the scaled Hamiltonian:
− 1
2m∆− κ α
m(−i∇) · A+ κ2(
α2
2mA2 +Hf) + V.
We introduce another scaling. Let
Hε(κ) = − 1
2m∆− κ2 α
m(−i∇) · A+ κ2(ε
α2
2mA2 +Hf) + V. (3.140)
The scaling (3.140) may reflect the interaction (−i∇) · A between the particle and
the field rather than the weak coupling limit. As will be seen below under (3.134)
the displacement operator does not disappear, and an effective potential appears
instead of effective mass. The scaling in (3.140) corresponds to the substitution:
ω → κ2ω, ϕ→ κ3ϕ, ε→ ε/κ2. (3.141)
90
We investigate the scaling limit of Hε(κ) as κ → ∞. Instead of energy renormal-
izations, in this scaling limit we need a mass renormalization. Under the scaling
(3.141), g and U are invariant. Let us define mren by
1
mren
=1
m+
α2(d−1d
)‖ϕ/ω‖2
m+ εα2(d−1d
)‖ϕ/ω‖2
1
m. (3.142)
Under the scaling (3.141), mren is scaled as
1
mren
→ 1
mren(κ)=
1
m+ κ2 α2
(d−1d
)‖ϕ/ω‖2
m+ εα2(d−1d
)‖ϕ/ω‖2
1
m.
We define the renormalized Hamiltonian by
Hren = − 1
2mren(κ)∆− κ2 α
m(−i∇) · A+ κ2
(α2
2mA2 +Hf
)+ V. (3.143)
Theorem 3.48 (Strong coupling limit) [Ara90, Hir93, Hir97, Hir02]Let V ∈L1
loc(Rd) be relatively bounded with respect to − 12m
∆ with a relative bound strictly
smaller than one. Suppose Assumption 3.4 and
∫ϕ2
ω3dk <∞. Then for z ∈ C \ R,
s-limκ→∞
(Hren−z)−1 = e−i(−i∇)·Π(
(− 1
2m∆ + Veff + g − z)−1 ⊗ PW
)ei(−i∇)·Π, (3.144)
where
Πµ =i√2
α
m
a∗(
ejµϕ
ω3/2)− a(
ejµϕ
ω3/2)
, (3.145)
PW = UWPΩU −1W , (3.146)
Veff(x) = V ∗ PC(x), (3.147)
PC(x) = (2πC)−d/2e−|x|2/(2C), (3.148)
C =1
2
(d− 1
d
)‖Qε/ω
3/2‖2. (3.149)
Proof: By the unitary transformation U we have
(Hren(κ)− z)−1 = U (− 1
2m∆ + κ2Hf + T−1V T + g − z)−1U −1. (3.150)
91
Scaling and renormalization Effective Hamiltonian
WCL 12m
(−i∇− ακA)2 + V + κ2Hf − κ2g − 12meff
∆ + V
SCL − 12mren(κ)
∆− κ2 αm
(−i∇)A+ κ2( α2
2mA2 +Hf) + V − 1
2m∆ + V ∗ PC + g
Figure 5: Scaling limits
By the abstract formula [Ara90] again, it has been established that
s-limκ→∞
(− 1
2m∆+κ2Hf +T
−1V T+g−z)−1 = (− 1
2m∆+(Ω, T−1V TΩ)F +g−z)−1⊗PΩ.
Hence we can see that
s-limκ→∞
(Hren(κ)− z)−1
= U
(− 1
2m∆ + (Ω, T−1V TΩ)F + g − z)−1 ⊗ PΩ
U −1
= e−i(−i∇)·Π(
(− 1
2m∆ + Veff + g − z)−1 ⊗ PW
)ei(−i∇)·Π.
Then the theorem is proven. 2
Potential Veff is called the effective potential . In the case of ε = 1 the effective
potential Veff and C are given by
Veff(x) = V ∗ PC , (3.151)
C = α2 1
2
(d− 1
d
)∫|ϕ(k)|2
meff(k)ω(k)3dk, (3.152)
and in the case of ε = 0,
Veff(x) = V ∗ PC , (3.153)
C = α2 1
2
(d− 1
d
)∫|ϕ(k)|2
m2ω(k)3dk. (3.154)
The strong coupling limit gives a mathematical interpretation of the Lamb shift
derived by (3.3). Namely the difference of the spectrum of − 12m
∆ +V and − 12m
∆ +
Veff approximately gives an interpretation of the Lamb shift. This was done in
[Wel48] and see historical review [Sch94, p.306,(7.4.20)]. See also [Ara90, Ara11].
92
3.10 Negative mass
We are interested in investigating the Hamiltonian with negative mass m < 0 from
mathematical point of view. This is of course an unphysical assumption. In this sec-
tion we suppose that −∑3
µ=1 α2‖λ/ω‖2 < m < 0. In this case W =
(W+ W−W− W+
)6∈
Sp. Then we define a new operators. Let us define X± by
X±ijf = W±ijf +1
2
1√ωeiµFµν(
√ωejνQ, f). (3.155)
Lemma 3.49 (Symplectic structure) Suppose Assumption 3.4. Then it follows
that
X∗+X+ −X∗−X− = 1l, (3.156)
X∗+X− −X∗−X+ = 0, (3.157)
X+X∗+ −X−X∗− = 1l, (3.158)
X−X∗+ −X+X∗− = 0. (3.159)
I.e., X =
(X+ X−X− X+
)∈ Sp.
Proof: Let ξ = (ξij)1≤i,j≤d−1 and ξij = 12
1√ωeiµFµν(
√ωejνQ, ·). Then we have (ξ∗)ij =
12
√ωeiµQ( 1√
ωejνFµν , ·). Thus X± = W± + ξ. We have
LHS(3.156) = 1l + ξ∗(W+ −W−) + (W ∗+ −W ∗
−)ξ
By (13) of Lemma 3.12 we see that
ξ∗(W+ −W−)f =1
2
√ωeiµQ(
1√ωekνFµν , (W+ −W−)kjf)
=1
2
√ωeiµQ(
1√ωekνFµν , e
ka
√ωT ∗ab
1√ωejbf)
=1
2
√ωeiµQ(ejbTabFaµ,
1√ωf) = 0.
We also see that
(W ∗+ −W ∗
−)ξf =1
2(√ωejνQ, f)(W ∗
+ −W ∗−)ik
1√ωekµFµν = 0.
93
Then (3.156) follows. Identity (3.157) is similarly proven. We have
LHS(3.158) = 1l + Z+ + (W+ξ∗ − W−ξ∗) + (ξW ∗
+ − ξW ∗−) + (ξξ∗ − ξξ∗).
We see that ξξ∗ − ξξ∗ = 0. By (12) and (15) of Lemma 3.12 we have
W−ξ∗f
=1
4eiµ
1√ωT ∗µνωdνaQ(
1√ωejbFab, f)− 1
4eiµ√ωT ∗µνdνaQ(
1√ωejbFab, f)
=1
4eiµ
1√ωT ∗µνωdνaQ(
1√ωejbFab, f)− 1
4eiµ√ωγ2Fµa(
1√ωejbFab, f)
=1
4eiµ
1√ωT ∗µνωdνaQ(
1√ωejbFab, f)− 1
4eiµ√ωγ2Fµa(
1√ωejbFab, f).
On the other hand
W+ξ∗f =
1
4eiµ
1√ωT ∗µνωdνaQ(
1√ωejbFab, f) +
1
4eiµ√ωγ2Fµa(
1√ωejbFab, f).
Hence
W+ξ∗ − W−ξ∗ =
1
2eiµ√ωγ2Fµa(
1√ωejbFab, ·)
follows. In a similar manner we have
ξW ∗−f = −1
4
1√ωeiµFµν(γ
2Fνb,√ωejbf) +
1
4
1√ωeiµFµν(dνa
√ωQ,√ωTab
1√ωejbf)
ξW ∗+f =
1
4
1√ωeiµFµν(γ
2Fνb,√ωejbf) +
1
4
1√ωeiµFµν(dνa
√ωQ,√ωTab
1√ωejbf).
Then
ξW ∗+ − ξW ∗
− =1
2
1√ωeiµγ
2Fµν(√ωejbFνb, ·).
Together with them we have
(W+ξ∗ − W−ξ∗) + (ξW ∗
+ − ξW ∗−) + (ξξ∗ − ξξ∗) = −Z+.
Then (3.158) follows. Finally we prove (3.159). We have
LHS(3.159) = Z+ + (W−ξ∗ − W+ξ
∗) + (ξW+ − ξW ∗−) + (ξξ∗ − ξξ∗).
94
By (12) and (15) of Lemma 3.12 we have
W+ξ∗f =
1
4eiµ
1√ωT ∗µνωdνaQ(
1√ωejbFab, f) +
1
4eiµ√ωT ∗µνdνaQ(
1√ωejbFab, f)
=1
4eiµ
1√ωT ∗µνωdνaQ(
1√ωejbFab, f) +
1
4eiµ√ωT ∗µνdνaQ(
1√ωejbFab, f)
=1
4eiµ
1√ωT ∗µνωdνaQ(
1√ωejbFab, f) +
1
4eiµ√ωγ2Fµa(
1√ωejbFab, f)
W−ξ∗f =
1
4eiµ
1√ωT ∗µνωdνaQ(
1√ωejbFab, f)− 1
4eiµ√ωγ2Fµa(
1√ωejbFab, f).
Hence
W−ξ∗ − W+ξ
∗ = −1
2γ2eiµ√ωFµa(
1√ωejbFab, f).
Similarly
ξW ∗−f = −1
4
1√ωeiµFµν(
√ωekνQ, e
ka(
1√ωTab√ω −√ωTab
1√ω
)ejbf)
= −1
4
1√ωeiµFµν(T
∗abdνaQ,
√ωejbf) +
1
4
1√ωeiµFµν(dνaQ, ωTab
1√ωejbf)
= −1
4
1√ωeiµFµν(T
∗abdνaQ,
√ωejbf) +
1
4
1√ωeiµFµν(dνaQ,ωTab
1√ωejbf)
= −1
4
1√ωeiµFµν(γ
2Fbν ,√ωejbf) +
1
4
1√ωeiµFµν(dνaQ,ωTab
1√ωejbf)
ξW+f =1
4
1√ωeiµFµν(γ
2Fbν ,√ωejbf) +
1
4
1√ωeiµFµν(dνaQ,ωTab
1√ωejbf).
Hence
ξW+ − ξW ∗− =
1
2
1√ωeiµγ
2Fµν(√ωejbFbν , ·)
and then
(W−ξ∗ − W+ξ
∗) + (ξW+ + ξW ∗−) + (ξξ∗ − ξξ∗) = −Z−.
Hence (3.159) follows. 2
Although
(W+ W−W− W+
)6∈ Sp, it is shown that B]
p(f, j) still satisfies canonical
commutation relations and adjoint relation. We notice that a] can not be realized
95
however in terms of B]p. For p ∈ Rd, we define
Cµ(p) = − 1
Epµ + EAa(Faµ)− Πa(Faµ),
Dµ(p) = − 1
Epµ + EAa(Faµ) + Πa(Faµ).
Both Cµ(p) and Dµ(p) are essentially self-adjoint.
Lemma 3.50 Suppose Assumption 3.4. Then it follows that
[Cµ(p), Dν(p)] = 2iE
γ2δµν , (3.160)
[Cµ(p), Cν(p)] = 0, (3.161)
[Dµ(p), Dν(p)] = 0, (3.162)
[Bp(f, j), Dν(p)] = [Bp(f, j), Cν(p)] = 0, (3.163)
[B∗p(f, j), Dν(p)] = [B∗p(f, j), Cν(p)] = 0. (3.164)
Proof: We can directly see that
[Cµ(p), Dν(p)] = 2E[Πa(Faµ), Ab(Fbν)] = 2Ei(dabFaµ, Fbν)
= 2Ei(Fbµ, Fbν) = 2iE
γ2δµν
by (16) of Lemma 3.12. Both (3.161) and (3.162) also follow from (16) of Lemma
3.12. Both (3.163) and (3.164) follow from (13) of Lemma 3.12. 2
Lemma 3.51 Suppose Assumption 3.4. Then it follows that
[H0, Cµ(p)] = iECµ(p), (3.165)
[H0, Dµ(p)] = −iEDµ(p). (3.166)
Proof: We have [Hf , Cµ(p)] = −iEΠa(Faµ)− iAa(ω2Faµ). Note that
ω2Faµ = αϕdaµ − E2Faµ.
Then
[Hf , Cµ(p)] = −iEΠa(Faµ)− iαAµ + iE2Aa(Faµ).
96
On the other hand we can see that
[Aν , Cµ(p)] = −i(ϕ, Fµν) = im
αδµν
by (14) of Lemma 3.12. Together with them we have
[H0, Cµ(p)] =1
m(pν − αAν)(−α)
mi
αδµν − iEΠa(Faµ)− iαAµ + iE2Aa(Faµ)
= −ipµ − iEΠa(Faµ) + iE2Aa(Faµ) = iECµ(p).
Then (3.165) follows. (3.166) is similarly proven. 2
Lemma 3.52 Suppose Assumption 3.4. Then it follows that
eitH0Cµ(p)e−itH0 = e−tECµ(p),
eitH0Dµ(p)e−itH0 = etEDµ(p).
Proof: Set Ct = etcECµ(p) and Ct = eitH0Cµ(p)e−itH0 . Then
d
dtCt =
d
dtCt = i[H0, Cµ(p)]
and C0 = C0. Then Ct = Ct follows. The equality eitH0Dµ(p)e−itH0 = etEDµ(p) is
similarly proven. 2
Theorem 3.53 (Time evolution of A) Suppose Assumption 3.4. Then for all
p ∈ Rd and real-valued f such that f ∈M0 ∩M−1/2 ∩M−1,
eitH0Aµe−itH0 =
1√2
B∗p(e
itωejν1√ωTνµf) +Bp(e
−itωejν1√ωTνµ
˜f)
+
α
meff
pν(dµνϕ
ω,f
ω) +
γ2
2E(Fµν , f)(e−tECν(p) + etEDν(p)). (3.167)
Proof: We shall show that
Aµ =1√2
B∗p(e
jν
1√ωTνµf) +Bp(e
jν
1√ωTνµ
˜f)
+
α
meff
pν(dµνϕ
ω,f
ω) +
γ2
2E(Fµν , f)(Cν(p) +Dν(p))
=1√2
B∗p(e
jν
1√ωTνµf) +Bp(e
jν
1√ωTνµ
˜f)
+
α
meff
pν(dµνϕ
ω,f
ω) +
(− γ
2
E2pν + γ2Aa(Faν)
)(Fµν , f). (3.168)
97
We see that
B∗p(ejν
1√ωTνµf)
= a(W−ijejν
1√ωTµν f , i) + a∗(W+ije
jν
1√ωTµν f , i)− (p · ejν
Q√2ω,
1
ωTµν f),
Bp(ejν
1√ωTνµf)
= a(W+ijejν
1√ωTµν
˜f, i) + a∗(W−ije
jν
1√ωTµν
˜f, i)− (p · ejν
Q√2ω,
1
ωTµν
˜f).
By using (3) and (7) of Lemma 3.12, we compute the sum of test function of the
creation operators as
W+ijejν
1√ωTµν f +W−ije
jν
1√ωTµν
˜f =
1√ωeiµ
˜f − γ2(Fµν ,
˜f)eiaFνa√ω
(3.169)
and that of the annihilation operators as
W−ijejν
1√ωTµν f +W+ije
jν
1√ωTµν
˜f =
1√ωeiµf − γ2(Fµν , f)
eiaFνa√ω. (3.170)
By (11) and (12) of Lemma 3.12 we have
− (p · ejνQ√2ω,
1
ωTµν f)− (p · ejν
Q√2ω,
1
ωTµν
˜f)
= −√
2pa
α
meff
(daνϕ
ω,f
ω)− γ2
E2(Faν , f)
. (3.171)
From (3.169)-(3.171), (3.168) follows. (3.167) is derived from Lemmas 3.22 and 3.52.
2
98
4 Binding and non-binding
4.1 Enhanced binding
Non-perturbative analysis of perturbation of eigenvalues embedded in the continu-
ous spectrum has been developed in the last decade and has been applied to the
mathematical rigorous analysis of Hamiltonians in quantum field theory. Among
other things, stability and instability of quantum mechanical particle coupled to
quantum fields have been investigated from mathematical point of view.
This section is the review of [HS01, Hir03, HSS11] and we also revise small errors
found in [HS01, Hir03, HSS11].
Atoms consist of charged particles and they are necessarily coupled to the quan-
tized radiation field. In the lowest approximation, this interaction can be ignored
and one is led to a Schrodinger operator of the form
Hp(m) = − 1
2m∆ + V (4.1)
for the particles only. Under suitable conditions on V the Schrodinger operator
has a state of the lowest energy, the ground state of the atom. There has been
renewed interest within mathematical physics to understand whether this ground
state persists when the coupling to the radiation field is included. We will investigate
here a related, but distinct problem.
In the non-relativistic approximation, the coupling to the radiation field is de-
scribed by the Pauli-Fierz Hamiltonian discussed in the previous section:
H =1
2m(−i∇− αA)2 + V +Hf (4.2)
acting on the Hilbert space H . In essence, V is short ranged and sufficiently
shallow. The problem of the existence of the ground state for H is usually regarded
as a stability property. One assumes that H has a ground state for α = 0, which
amounts to the existence of a ground state for Hp(m) and proves that H has also
a ground state for α 6= 0. It is then necessarily unique, since e−tH has a positivity
improving kernel in a suitable function space.
In contrast we assume that H has no ground state for α = 0. In fact, this will
be the case for a sufficiently shallow V in the space dimension three. We expect
99
Figure 6: Enhanced binding by effective mass
the interaction with the quantized radiation field to enhance binding. The non-
binding potential should become binding at a sufficiently strong coupling strength.
The enhanced binding is studied in e.g., [AK03, BLV05, BV04, CEH04, CVV03,
HVV03, HS08, HS12, HS01, HSS11].
The physical reasoning behind such a result is simple. As the particle binds
photons it acquires an effective mass meff = m+α2(d−1d
)‖ϕ/ω‖2 which is increasing
in |α| (Figure 6). Roughly speaking, H may be replaced by
Heff = − 1
2meff
∆ + V, (4.3)
which binds for sufficiently strong α. Indeed we can see that Heff can be derived
through the weak coupling limit of H in Section 3.9.1.
Next let us consider a transition from unbinding to binding as the mass m is
increased (Figure 7). More precisely, there is some critical mass, mc, such that
Hp(m) has no ground state for 0 < m < mc and a unique ground state for mc < m.
In fact, the critical mass is given by
1
2mc
=∥∥|V |1/2 (−∆)−1 |V |1/2
∥∥ . (4.4)
On a heuristic level, through the dressing by photons the particle becomes effectively
more heavy, which means that there is critical mass mc(α) for the existence of a
ground state. mc(α) is expected to be decreasing as a function of α with mc(0) = mc.
In particular, for fixed m < mc, there should be an unbinding-binding transition as
the coupling α is increased. In case m > mc more general techniques are available
and the existence of a unique ground state for the full Hamiltonian H is proven in
100
Figure 7: Transition from unbinding to binding
[BFS99, GLL01]. The heuristic picture also asserts that the full Hamiltonian has a
regime of couplings with no ground state.
4.2 Absence of ground state
The unbinding for the Schrodinger operator Hp(m) = − 12meff
∆ +V is proven by the
Birman-Schwinger principle. Formally one has
Hp(m) =1
2m(−∆)1/2
(1l + 2m(−∆)−1/2V (−∆)−1/2
)(−∆)1/2.
If m is sufficiently small, then 2m(−∆)−1/2V (−∆)−1/2 is a strict contraction. Hence
the operator 1l + 2m(−∆)−1/2V (−∆)−1/2 has a bounded inverse and Hp(m) has no
eigenvalue in (−∞, 0]. More precisely the Birman-Schwinger principle states that
dim1l[ 12m
,∞)(V1/2(−∆)−1V 1/2) ≥ dim1l(−∞,0](Hp(m)). (4.5)
For small m the left hand side equals 0 and thus Hp(m) has no eigenvalues in
(−∞, 0]. Our approach will be to generalize (4.5) to the Pauli-Fierz model with the
dipole approximation. We already see that H can be transformed by U and one
arrives at U −1HU = H0(α) +W + g as the sum of the free Hamiltonian
H0(α) = − 1
2meff
∆ +Hf , (4.6)
101
involving the effective mass of the dressed particle, the transformed interaction W =
T−1V T , and the global energy shift g. Effective mass meff is an increasing function
of α.
Let h0 = −12∆. We assume that V ∈ L1
loc(Rd) and V is relatively form-bounded
with respect to h0 with relative bound a < 1, i.e., D(|V |1/2) ⊃ D(h1/20 ) and
||V |1/2ϕ‖2 ≤ a‖h1/20 ϕ‖2 + b‖ϕ‖2, ϕ ∈ D(h
1/20 ), (4.7)
with some b > 0. Under 4.7 the operators RE = (h0 − E)−1/2 |V |1/2 for E < 0 are
densely defined. From (4.7) it follows that R∗E = |V |1/2(h0−E)−1/2 is bounded and
thus RE is closable. We denote its closure by the same symbol. Let
KE = R∗ERE. (4.8)
Then KE (E < 0) is a bounded, positive self-adjoint operator and it holds
KEf = |V |1/2 (h0 − E)−1 |V |1/2f, f ∈ C∞0 (Rd).
Now let us consider the case E = 0. Let R0 = h−1/20 |V |1/2. The self-adjoint
operator h−1/20 has the integral kernel h
−1/20 (x, y) =
ad|x− y|d−1
for d ≥ 3, where
ad =√
2π(d−1)/2/Γ(d−12
). It holds that∣∣∣(h−1/20 g, |V |1/2f)
∣∣∣ ≤ ad‖g‖2‖|V |1/2f‖2d/(d+2)
for f, g ∈ C∞0 (R3) by the Hardy-Littlewood-Sobolev inequality. Since f ∈ C∞0 (R3)
and V ∈ L1loc(R3), one concludes ‖|V |1/2f‖2d/(d+2) < ∞. Thus |V |1/2f ∈ D(h
−1/20 )
and R0 is densely defined. Since V is relatively form-bounded with respect to h0,
R∗0 is also densely defined, and R0 is closable. We denote the closure by the same
symbol. We define
K0 = R∗0R0. (4.9)
Next let us introduce assumptions on the external potential V .
Assumption 4.1 V satisfies that (1) V ≤ 0 and (2) R0 is compact.
Lemma 4.2 Suppose Assumption 4.1. Then
(1) RE, R∗E and KE (E ≤ 0) are compact.
102
(2) ‖KE‖ is continuous and monotonously increasing in E ≤ 0 and it holds that
limE→−∞
‖KE‖ = 0 and limE↑0 ‖KE‖ = ‖K0‖.
Proof: Under (2) of Assumption 4.1, R∗0 and K0 are compact. Since
(f,KEf) ≤ (f,K0f), f ∈ C∞0 (Rd), (4.10)
extends to f ∈ L2(R3), KE, RE and R∗E are also compact. Thus (1) is proven.
We will prove (2). It is clear from (4.10) that KE is monotonously increasing in
E. Since R0 is bounded, (4.10) holds on L2(Rd) and
KE = R∗0((h0 − E)−1h0
)R0 (4.11)
for E ≤ 0. From this one concludes that ‖KE −KE′‖ ≤ ‖K0‖ |E−E′|
|E′| for E,E ′ < 0.
Hence ‖KE‖ is continuous in E < 0. We have to prove the left continuity at E = 0.
Since ‖KE‖ ≤ ‖K0‖ (E < 0), one has lim supE↑0 ‖KE‖ ≤ ‖K0‖. By (4.11) we see
that K0 = s- limE↑0KE and
‖K0f‖ = limE↑0‖KEf‖ ≤
(lim infE↑0
‖KE‖)‖f‖, f ∈ L2(Rd).
Hence we have ‖K0‖ ≤ lim infE↑0 ‖KE‖ and limE↑0 ‖KE‖ = ‖K0‖. It remains to
prove that limE→−∞ ‖KE‖ = 0. Since R∗0 is compact, for any ε > 0, there exists a
finite rank operator Tε =∑n
k=1(ϕk, ·)ψk such that n = n(ε) < ∞, ϕk, ψk ∈ L2(Rd)
and ‖R∗0 − Tε‖ < ε. Then it holds that ‖KE‖ ≤ (ε+ ‖Tεh0(h0 − E)−1‖) ‖R0‖. For
any f ∈ L2(Rd), we have
‖Tεh0(h0 − E)−1f‖ ≤
(n∑k=1
‖h0(h0 − E)−1ϕk‖‖ψk‖
)‖f‖
and limE→−∞ ‖Tεh0(h0 − E)−1‖ = 0, which completes (2). 2
Let Hp(m) be in (4.1). By (2) of Lemma 4.2, we have
limE→−∞
‖|V |1/2(h0 − E)−1/2‖ = 0.
Therefore V is infinitesimally form bounded with respect to h0 and Hp(m) is the
self-adjoint operator associated with the quadratic form
f, g 7→ 1
m(h
1/20 f, h
1/20 g) + (|V |1/2f, |V |1/2g)
103
for f, g ∈ D(h1/20 ). Note that the domain D(Hp(m)) is independent of m.
Under (2) of Assumption 4.1, the essential spectrum of Hp(m) coincides with
that of − 12m
∆, hence σess(Hp(m)) = [0,∞). Next we will estimate the spectrum
of Hp(m) contained in (−∞, 0]. Let 1l(O)(T ), O ⊂ R, be the spectral resolution of
self-adjoint operator T and set NO(T ) = dim Ran1lO(T ). The Birman-Schwinger
principle states that
(E < 0) N(−∞, Em
] (Hp(m)) = N[ 1m,∞)(KE),
(E = 0) N(−∞,0] (Hp(m)) ≤ N[ 1m,∞)(K0).
(4.12)
Now let us define the constant mc by the inverse of the operator norm of K0,
mc = ‖K0‖−1. (4.13)
Lemma 4.3 Suppose Assumption 4.1.
(1) If m < mc, then N(−∞,0](Hp(m)) = 0.
(2) If m > mc, then N(−∞,0](Hp(m)) ≥ 1.
Proof: It is immediate to see (1) by the Birman-Schwinger principle (4.12). Suppose
m > mc. Then, using the continuity and monotonicity of E → ‖KE‖, see Lemma
4.2, there exists ε > 0 such that mc < ‖K−ε‖−1 ≤ m. Since K−ε is positive and
compact, ‖K−ε‖ ∈ σp(K−ε) follows and hence N[ 1m,∞)(K−ε) ≥ 1. Therefore (2)
follows again from the Birman-Schwinger principle. 2
By Lemma 4.3, the critical mass at zero coupling is mc(0) = mc. In the case
m > mc, by the proof of Lemma 4.3 one concludes that the bottom of the spectrum
of Hp(m) is strictly negative. For ε > 0 we set mε = ‖K−ε‖−1.
Corollary 4.4 Suppose Assumption 4.1 and m > mε. Then
inf σ (Hp(m)) ≤ −εm. (4.14)
Proof: The Birman-Schwinger principle states that 1 ≤ N(−∞,− εm
] (Hp(m)), since
1/m < ‖K−ε‖, which implies the corollary. 2
We extend the Birman-Schwinger type estimate to the Pauli-Fierz Hamiltonian.
104
Lemma 4.5 Suppose Assumption 4.1. If m < mc, then the zero coupling Hamilto-
nian Hp(m) +Hf has no ground state.
Proof: Since the Fock vacuum Ω is the ground state of Hf , Hp(m)+Hf has a ground
state if and only if Hp(m) has a ground state. But Hp(m) has no ground state by
Lemma 4.3. Therefore Hp(m) +Hf has no ground state. 2
From now on we discuss U −1HU with α 6= 0. We have
U −1HU = H0(α) +W + g, (4.15)
where H0(α) = − 1
2meff
∆ +Hf and W = T−1V T and T is given in (3.31).
Theorem 4.6 (Absence of ground state) [HSS11] Suppose Assumptions 3.4 and
4.1. If meff < mc, then H has no ground state.
Proof: Since g is a constant, we prove the absence of ground state of H0(α) + W .
Since V is negative, so is W . Hence inf σ(H0(α) +W ) ≤ inf σ(H0(α)) = 0. Then it
suffices to show that H0(α) + W has no eigenvalues in (−∞, 0]. Let E ∈ (−∞, 0]
and set
KE = |W |1/2(H0(α)− E)−1|W |1/2, (4.16)
where |W |1/2 is defined by the functional calculus. We shall prove now that if
H0(α) + W has eigenvalue E ∈ (−∞, 0], then KE has eigenvalue 1. Suppose that
(H0(α) + W − E)ϕ = 0 and ϕ 6= 0, then KE|W |1/2ϕ = |W |1/2ϕ holds. Moreover
if |W |1/2ϕ = 0, then Wϕ = 0 and hence (H0(α) − E)ϕ = 0, but H0(α) has no
eigenvalue by Lemma 4.5. Then |W |1/2ϕ 6= 0 is concluded and KE has eigenvalue 1.
Then it is sufficient to see ‖KE‖ < 1 to show that H0(α) +W has no eigenvalues in
(−∞, 0]. Notice that − 12meff
∆ and T commute, and∥∥∥(−∆)1/2 (H0(α)− E)−1 (−∆)1/2∥∥∥ ≤ 2meff .
Then we have
‖KE‖ ≤
∥∥∥∥∥|V |1/2(− 1
2meff
∆
)−1/2∥∥∥∥∥
2
= meff‖K0‖ =meff
mc
< 1
and the proof is complete. 2
105
Now we give examples of potentials V satisfying Assumption 4.1. The self-adjoint
operator h−10 has the integral kernel
h−10 (x, y) =
bd|x− y|d−2
, d ≥ 3,
with bd = 2Γ(d2− 1)/π
d2−2. It holds that
(f,K0f) =
∫dx
∫dyf(x)K0(x, y)f(y), (4.17)
where
K0(x, y) = bd|V (x)|1/2|V (y)|1/2
|x− y|d−2, d ≥ 3, (4.18)
is the integral kernel of operator K0. We recall that the Rollnik class R of potentials
is defined by
R =
V∣∣∣ ∫
Rddx
∫Rddy|V (x)V (y)||x− y|2
<∞.
Let d = 3. By the Hardy-Littlewood-Sobolev inequality, R ⊃ Lp(R3)∩Lr(R3) with
1/p+ 1/r = 4/3. In particular, L3/2(R3) ⊂ R.
Example 4.7 (d = 3 and Rollnik class) Let d = 3. Suppose that V is negative
and V ∈ R. Then K0 ∈ L2(R3×R3). Hence K0 is Hilbert-Schmidt and Assumption
4.1 is satisfied.
The example can be extended to dimensions d ≥ 3.
Example 4.8 (d ≥ 3 and V ∈ Ld/2(Rd)) Let Lpw(Rd) be the set of Lebesgue
measurable function u such that supβ>0 β∣∣x ∈ Rd‖u(x) > β
∣∣1/pL
<∞, where |E|Ldenotes the Lebesgue measure of E ⊂ Rd. Let g ∈ Lp(Rd) and u ∈ Lpw(Rd) for
2 < p <∞. Define the operator Bu,g by
Bu,gh = (2π)−d/2∫eikxu(k)g(x)h(x)dx.
It is shown in [Cwi77, Theorem, p.97] that Bu,g is a compact operator on L2(Rd).
It is known that u(k) = 2|k|−1 ∈ Ldw(Rd) for d ≥ 3. Let F denote Fourier transform
on L2(Rd), and suppose that V ∈ Ld/2(Rd). Then Bu,|V |1/2 is compact on L2(Rd)
and then R∗0 = FBu,V 1/2F−1 is compact. Thus R0 is also compact.
106
Assume that V ∈ Ld/2(Rd). Let us now see the critical mass of zero coupling
mc = m0. By the Hardy-Littlewood-Sobolev inequality, we have
|(f,K0f)| ≤ DV ‖f‖22, (4.19)
whereDV =√
2πΓ(d
2− 1)
Γ(d2
+ 1)
(Γ(d)
Γ(d2)
)2/d
‖V ‖2d/2, (4.19) is proved by Lieb [Lie83]. Then
‖K0‖ ≤ DV . By this bound we have mc ≥ D−1V . In particular in the case of d = 3,
mc ≥3√
2π2/345/3‖V ‖−2
3/2. (4.20)
4.3 Existence of ground state
In this section we investigate the existence of ground state of H for sufficiently
large |α|. Let us define the Pauli-Fierz Hamiltonian with scaled external potential
Vκ(x) = V (x/κ)/κ2 by1
2m(−i∇− αA)2 + Vκ +Hf . (4.21)
We also define H(κ) by H with a] replaced by κa]. Then
H(κ) =1
2m(−i∇− καA)2 + V + κ2Hf . (4.22)
We can see the unitary equivalence:
κ−2H(κ) ∼=1
2m(−i∇− αA)2 + Vκ +Hf .
Then H(κ) has a ground state if and only if (4.21) has a ground state. We further-
more introduce assumptions on the external potential V and ultraviolet cutoff ϕ.
Recall that Q(k) = αϕ(k)/meff(k).
Assumption 4.9 The external potential V and the ultraviolet cutoff ϕ satisfy:
(1) V ∈ C1(Rd) and ∇V ∈ L∞(Rd),
(2) ϕ/ω5/2 ∈ L2(Rd),
107
(3) supα ‖Q/ωn/2‖ <∞, n = 3, 4, 511.
Example 4.10 We give an example of ultraviolet cutoff satisfying both of Assump-
tion 3.4 and Assumption 4.9 (2) and (3). Suppose that d = 3 and ϕ(k) = 1l[λ,Λ](|k|)is the sharp cutoff function. See Example 3.10. Then
|meff(k)| ≥ α2 4π2
31l[λ,Λ](ω(k))
√ω(k).
We have
‖Q/ωn/2‖2 ≤ 1
α2
(3
4π2
)2 ∫λ≤|k|≤Λ
1
ω(k)n+1dk.
In particular it follows that limα→∞ ‖Q/ωn/2‖ = 0.
We drop g for instance. We reset
H(κ) = Heff + κ2Hf + δVκ. (4.23)
In Theorem 3.47 we show that H(κ) converges to Heff as κ → ∞ in some sense.
It suggests that H(κ) with sufficiently large κ has a ground state if Heff does. Let
m < mc and ε > 0. We define
αε =
((d− 1
d
)‖ϕ/ω‖2
)−1/2√mε −m, ε > 0, (4.24)
α0 =
((d− 1
d
)‖ϕ/ω‖2
)−1/2√mc −m, (4.25)
where we recall that mε = ‖K−ε‖−1 for ε ≥ 0. Note that
(1) |α| < α0 if and only if meff < mc;
(2) |α| > αε if and only if meff > mε.
Note that α0 < αε because of mε > mc. Since limε↓0mε = mc, it holds that
limε↓0 αε = α0. We note that for |α| > αε, Heff has a ground state with negative
ground state energy.
11[HS01, Theorem 4.14] is incorrect. The effective mass meff in [HS01, Theorem 4.14] should bechanged to meff(k), and we need assumption (3) to show the enhanced binding.
108
4.3.1 Massive case
We introduce an artificial mass of photon, ε > 0, and define
Hε(κ) = Heff + δVκ + κ2Hεf ,
where
Hεf = Hf + εN =
∫(ω(k) + ε)a∗(k, j)a(k, j)dk.
Using a momentum lattice approximation we will prove that Hε(κ) has a ground
state. Let Γ(l, a), l = (l1, · · · , ld) ∈ Zd, a > 0, be the momentum lattice with spacing
1/a, i.e., Γ(l, a) = [ l1a, (l1+1)
a)× · · · × [ ld
a, (ld+1)
a) and
χΓ(l,a)(k) =
0, k 6∈ Γ(l, a),ad/2, k ∈ Γ(l, a).
For L > 0 we define the momentum-lattice-approximated Hamiltonian by
Hεa,L(κ) = Heff + κ2Hε′
f + δV ′κ, (4.26)
where Hε′
f and δV ′κ are momentum-lattice-approximated operators given by
Hε′
f = Hεf,a,L =
∫ ∑|l|≤L
χΓ(l,a)(k)(ω(l) + ε)
a∗(k, j)a(k, j)dk,
δV ′κ = δVκ,a,L = V (·+Ka,L/κ)− V
and Ka,L = (Ka,L,1, · · · , Ka,L,d) is the column of the field operator defined by
Ka,L,µ =1√2
∫ ∑|l|≤L
χΓ(l,a)(k)(%µ(l, j)a∗(k, j) + %µ(l, j)a(k, j))dk.
Here we set %µ(k, j) = ejµ(k)Q(k)/ω(k)3/2. We can show that
‖Ka,L,µΨ‖ ≤ C
∥∥∥∥∥∥∑|l|≤L
χΓ(l,a)Q(l)√ωω(l)3/2
∥∥∥∥∥∥+
∥∥∥∥∥∥∑|l|≤L
χΓ(l,a)Q(l)
ω(l)3/2
∥∥∥∥∥∥ (‖(Hε′
f )1/2Ψ‖+ ‖Ψ‖)
(4.27)
with some constant C. Here we used the bound
c1‖(Hε′
f )1/2Ψ‖ ≤ ‖(Hεf )1/2Ψ‖ ≤ c2‖(Hε′
f )1/2Ψ‖
with some constants c1 and c2.
109
Lemma 4.11 It follows that limL→∞
lima→∞
Hεa,L(κ) = Hε(κ) in the uniform resolvent
sense.
Proof: It can be seen that there exists a constant ca,L such that
‖(Hε′
f −Hεf )Ψ‖ ≤ ca,L‖Hε
f Ψ‖
and limL→∞
lima→∞
ca,L = 0. Moreover
‖(δV ′κ − δVκ)Ψ‖ = ‖(V (·+Ka,L/κ)− V (·+K/κ))Ψ‖
≤ 1
κ‖∇µV ‖∞‖(Kµ −Ka,L,µ)Ψ‖.
Since ‖(Kµ −Ka,L,µ)Ψ‖ ≤ c′a,L(‖(Hεf )1/2Ψ‖+ ‖Ψ‖), where
c′a,L = C
∥∥∥∥∥∥ 1√ω
Q
ω3/2−∑|l|≤L
χΓ(l,a)Q(l)
ω(l)3/2
∥∥∥∥∥∥+
∥∥∥∥∥∥ Q
ω3/2−∑|l|≤L
χΓ(l,a)Q(l)
ω(l)3/2
∥∥∥∥∥∥
with some constant C, and c′a,L satisfies that limL→∞
lima→∞
c′a,L = 0, we have
‖(Hε(κ)− z)−1Ψ− (Hεa,L(κ)− z)−1Ψ‖
≤ ‖(Hεa,L(κ)− z)−1‖‖(Hε
a,L(κ)−Hε(κ))(Hε(κ)− z)−1Ψ‖
≤(maxµ ‖∇µV ‖∞)c′a,L
|Imz|(‖(Hε
f )1/2(Hε(κ)− z)−1Ψ‖+ ‖(Hε(κ)− z)−1Ψ‖)
+ca,L|Imz|
‖(Hεf )1/2(Hε(κ)− z)−1Ψ‖.
Since ‖Hεf (Hε(κ)− z)−1Ψ‖ ≤ C ′‖Ψ‖ with some constant C ′, we have
‖(Hε(κ)− z)−1Ψ− (Hεa,L(κ)− z)−1Ψ‖ ≤ c′′a,L‖Ψ‖
with c′′a,L such that limL→∞ lima→∞ c′′a,L = 0. Hence the lemma follows. 2
Let f ∈ L2(Rd). We identify
`2(Zd) 3 f(l)l∈Zd ∼= ad/2∑l∈Zd
f(l)χΓ(l,a)(·) ∈ L2(Rd).
110
By this identification we regard `2 = `2(Zd) as the subspace of L2(Rd). Let
Ha = L2(Rd)⊗F (`2 ⊗ Cd−1), (4.28)
K = ⊕∞n=1F(n)(`2⊥ ⊗ Cd−1). (4.29)
Then the following fundamental identification follows:
H = L2(Rd)⊗F (L2(Rd × 1, ..., d− 1))∼= L2(Rd)⊗F (L2(Rd)⊗ Cd−1)
∼= L2(Rd)⊗F ([`2 ⊕ `2⊥]⊗ Cd−1)
∼= L2(Rd)⊗ [F (`2 ⊗ Cd−1)⊗F (`2⊥ ⊗ Cd−1)]
∼= Ha ⊗F (`2⊥ ⊗ Cd−1)
= Ha ⊗ (K ⊕ C)
∼= (Ha ⊗K )⊕Ha.
We have
H ∼= (Ha ⊗K )⊕Ha.
In particular we can see that
H ⊥a∼= Ha ⊗K (4.30)
and that Hεa,L(κ) is reduced by Ha. We set
K = Hεa,L(κ)
⌈Ha,
K⊥ = Hεa,L(κ)
⌈H ⊥a.
Then
Hεa,L(κ) = K⊥ ⊕K.
We can immediately see the lemma below:
Lemma 4.12 Under the identification (4.30), we have
K⊥ ∼= K ⊗ 1l + 1l⊗ κ2Hε′
f
⌈K.
In particular infσ(K⊥) ≥ infσ(K) + ε.
111
In what follows we estimate the spectrum of K.
Lemma 4.13 Let Ψ ∈ D(−∆) ∩D(Hf1/2). Then
(1) Ψ ∈ D(δVκ) and
‖δVκΨ‖ ≤ θκ(‖(Hεf )1/2Ψ‖+ ‖Ψ‖), (4.31)
where θκ = 1κC‖∇V ‖∞(‖Q/ω2‖+ ‖Q/ω3/2‖) with some constant C,
(2) Ψ ∈ D(δV ′κ) and
‖δV ′κΨ‖ ≤ θ′κ(‖(Hε′
f )1/2Ψ‖+ ‖Ψ‖), (4.32)
where
θ′κ = θκ,a,L =1
κC ′‖∇V ‖∞
∥∥∥∥∥∥ 1√ω
∑|l|≤L
χΓ(l,a)Q(l)
ω(l)3/2
∥∥∥∥∥∥+
∥∥∥∥∥∥∑|l|≤L
χΓ(l,a)Q(l)
ω(l)3/2
∥∥∥∥∥∥
with some constant C ′.
Proof: We have ‖δVκΨ‖ ≤ 1κ‖∇µV ‖∞‖Kµ,a,LΨ‖. Then (4.31) follows. (4.32) is
similarly proven. 2
Lemma 4.14 It follows that
infσ(Hε(κ)) ≤ infσ(Heff) +3θκ2,
infσ(Hεa,L(κ)dHa) ≤ infσ(Heff) +
3θ′κ2.
Proof: We write A ≤ B, if D(B) ⊂ D(A) and (ψ,Aψ) ≤ (ψ,Bψ) for ψ ∈ D(B). We
have
|(Ψ, δVκΨ)| ≤ θκ
‖Ψ‖(‖Hε
f1/2Ψ‖+ ‖Ψ‖)
≤ (Ψ, θκ(
3
2+
1
2Hε
f )Ψ).
Thus we can get the bound
− θκ(
1
2Hε
f +3
2
)≤ δVκ ≤ θκ
(1
2Hε
f +3
2
). (4.33)
112
Hence for f ∈ C∞0 (Rd),
infσ(Hε(κ)) ≤ (f ⊗ Ω, Hε(κ)f ⊗ Ω) ≤ (f, (Heff +3
2θκ)f).
In particular, since C∞0 (Rd) is a core of Heff , we have
infσ(Hε(κ)) ≤ infσ(Heff) +3θκ2.
Similarly we have
− θ′κ(
1
2Hε′
f +3
2
)≤ δV ′κ ≤ θ′κ
(1
2Hε′
f +3
2
)(4.34)
and hence
infσ(Hεa,L(κ)) ≤ infσ(Heff) +
3θ′κ2.
Then the lemma follows. 2
We set Σ = infσ(Heff) and Heff = Heff−Σ. Suppose |α| > αε. Since meff > mε >
mε/2,
Σ ≤ infσ(Hp(mε)) ≤ −ε
2mε
(4.35)
by Corollary 4.4. In particular
|Σ| > 0. (4.36)
For a self-adjoint operator M , the spectral projection of M on a Borel set B ⊂ R is
denoted by EMB .
Lemma 4.15 Suppose |α| > αε. Let a, L and κ be sufficiently large such that
min|Σ|/3, 2κ2 > θ′κ. Then for ε such that |Σ| > 3θ′κ + ε, we have
K − infσ(K)− ε ≥ EHeff
[0,|Σ|) ⊗(
(κ2 − θ′κ2
)Hε′
f − 3θ′κ − ε).
113
Proof: We directly see by Lemma 4.14 that
K − infσ(K)− ε= Heff + δV ′κ + κ2Hε′
f − infσ(H)− ε
≥ Heff + δV ′κ + κ2Hε′
f −3
2θ′κ − Σ− ε
≥ Heff + (κ2 − θ
2)Hε′
f −3
2θ′κ −
3
2θ′κ − Σ− ε
= Heff + (κ2 − θ′κ2
)Hε′
f − 3θ′κ − ε
≥ |Σ|EHeff
[|Σ|,∞) ⊗ 1l− θ′′κ(EHeff
[0,|Σ|) + EHeff
[|Σ|,∞))⊗ 1l + (κ2 − θ′κ2
)(EHeff
[0,|Σ|) + EHeff
[|Σ|,∞))⊗Hε′
f ,
where θ′′κ = 3θ′κ + ε. Then
K − infσ(K)− ε
≥ (|Σ| − θ′′κ)EHeff
[|Σ|,∞) ⊗ 1 + (κ2 − θ′κ2
)EHeff
[|Σ|,∞) ⊗Hε′
f
+ EHeff
[0,|Σ|) ⊗(
(κ2 − θ′κ2
)Hε′
f − θ′′κ).
Since |Σ| − θ′′κ = |Σ| − 3θ′κ − ε > 0 and κ2 − θ′κ2> 0 by the assumption, we have
K − infσ(K)− ε ≥ EHeff
[0,|Σ|) ⊗(
(κ2 − θ′κ2
)Hε′
f − θ′′κ).
Thus the lemma follows. 2
Set T = K − infσ(K)− ε as an operator in Ha. Define Ha(+) = ET[0,∞)Ha and
Ha(−) = ET[−ε,0)Ha.
Lemma 4.16 Suppose |α| > αε and that min|Σ|/3, 2κ2 > θ′κ. Then for ε such
that |Σ| > 3θ′κ + ε, T dHa(−) has a purely discrete spectrum, i.e.,
σ(K) ∩ [infσ(K), infσ(K) + ε) ⊂ σdisc(K).
Proof: Let φnn be a complete orthonormal system of Ha(−) and ψmm that of
Ha(+). We see that by Lemma 4.15,
0 ≥ tr T dHa(−) =∑n
(φn, Tφn) ≥∑n
(φn, T′φn),
114
where T ′ = EHeff
[0,|Σ|) ⊗(
(κ2 − θ′κ2
)Hε′
f − 3θ′κ − ε)
. Set T ′− = T ′ET ′
(−∞,0). Then
0 ≥ tr T dHa(−) ≥∑n
(φn, T′−φn) ≥
∑n
(φn, T′−φn) +
∑m
(ψm, T′−ψm) = trT ′−.
Hence we obtain that∣∣∣tr T dHa(−)
∣∣∣ ≤ ∣∣trT ′−∣∣ = trEHeff
[0,|Σ|) ×∣∣∣∣tr((κ2 − θ′κ
2) Hε′
f
⌈Ha
− 3θ′κ − ε)−
∣∣∣∣ ,where (· · ·)− denotes the negative part of (· · ·). Since σ(Hε′
f
⌈Ha
) = σdisc(Hε′
f
⌈Ha
)
and∣∣∣trEHeff
[0,|Σ|)
∣∣∣ <∞, it follows that∣∣∣tr T dHa(−)
∣∣∣ <∞. Thus the lemma follows. 2
Lemma 4.17 Suppose that min|Σ|/3, 2κ2 > θ′κ. Then for ε such that |Σ| >3θ′κ + ε, it follows that
σ(Hεa,L(κ)) ∩ [infσ(Hε
a,L(κ)), infσ(Hεa,L(κ)) + ε) ⊂ σdisc(H
εa,L(κ)).
Proof: We have by Lemmas 4.12 and 4.16,
σ(Hεa,L(κ)) = σ(K⊥) ∪ σ(K),
σ(K⊥) ⊂ [infσ(K) + ε,∞),
σ(K) ∩ [inf σ(K), inf σ(K) + ε) ⊂ σdisc(K).
Notice that infσ(K) = infσ(Hεa,L(κ)). Then the lemma follows. 2
Now we can show the existence of ground state of massive Hamiltonian Hε(κ).
Lemma 4.18 Suppose |α| > αε and that min|Σ|/3, 2κ2 > θκ. Then for ε such
that |Σ| > 3θκ + ε,
σ(Hε(κ)) ∩ [infσ(Hε(κ)), infσ(Hε(κ)) + ε) ⊂ σdisc(Hε(κ)).
In particular Hε(κ) has a ground state.
Proof: Note that limL→∞
lima→∞
θ′κ = θκ. Then by Lemmas 4.11 and 4.17, the lemma
follows. 2
See Figure 8 for the spectrum of massive Pauli-Fierz Hamiltonian.
115
Figure 8: Spectrum of massive Hamiltonian
4.3.2 Massless case
A ground state of Hε is denoted by Ψε.
Lemma 4.19 Suppose |α| > αε, Assumptions 3.4 and 4.9, and that min|Σ|/3, 2 >θ. Then for ε such that |Σ| > 3θκ + ε,
‖N1/2Ψε‖‖Ψε‖
≤ 1
κ2C‖Q/ω5/2‖(max
µ‖∇µV ‖∞) (4.37)
with some constant C.
Proof: We set E = infσ(Hε(κ)). Since
[Hε(κ), a(k, j)] = −(ω(k) + ε)a(k, j) + [δVκ, a(k, j)],
we have
Hε(κ)a(k, j)Ψε =− (ω(k) + ε)a(k, j)Ψε + Ea(k, j)Ψε + [δVκ, a(k, j)]Ψε.
Hence we derive that
(Hε(κ)− E + ω(k) + ε)a(k, j)Ψε = [δVκ, a(k, j)]Ψε (4.38)
and
[δVκ, a(k, j)] =
[V (·+ 1
κK), a(k, j)
]= T−1
κ
[V, Tκa(k, j)T−1
κ
]Tκ.
Since
Tκa(k, j)T−1κ = a(k, j)− 1
κ
i√2
(−i∇ν)%ν(k, j),
it follows that
[δVκ, a(k, j)] = T−1κ
[V,− i√
2κ(−i∇ν)%
ν(k, j)
]Tκ
=1
κT−1κ
(1√2
(∇νV )%ν(k, j)
)Tκ.
116
Thus we obtain the pull-through formula:
a(k, j)Ψε =1
κ(Hε(κ)− E + ω(k) + ε)−1T−1
κ
(1√2
(∇νV )%ν(k, j)
)TκΨε. (4.39)
Using identity (4.39) we see that
‖N1/2Ψε‖2 =d−1∑j=1
∫‖a(k, j)Ψε‖2dk
=1
2κ2
d−1∑j=1
∫ ∥∥(Hε(κ)− E + ω(k) + ε)−1T−1κ (∇νV )%ν(k, j)TκΨε
∥∥2dk
We then estimate as
‖N1/2Ψε‖2
‖Ψε‖2≤ 1
2κ2
d−1∑j=1
∫ (1
ω(k)‖∇νV ‖∞
)2
|%ν(k, j)|2 dk
≤ 1
κ2C(max
µ‖∇µV ‖∞)2‖Q/ω5/2‖2.
Hence the lemma follows. 2
Lemma 4.20 Suppose |α| > αε, Assumptions 3.4 and 4.9. Let PΩ be the projection
onto αΩ | α ∈ C and QΩ = EHeff
[Σ+δ,∞) ⊗ PΩ with some δ > 0 such that δ > 32θκ.
Suppose that min|Σ|/3, 2κ2 > θκ. Then for ε such that |Σ| > 3θκ + ε, it follows
that
‖QΩΨε‖‖Ψε‖
≤
√θκ
δ − 32θκ. (4.40)
Proof: Since (Ψε, QΩ(Hε(κ)− infσ(Hε(κ)))Ψε) = 0, we have
(Ψε, QΩ(Heff − infσ(Hε(κ)))Ψε) = −(Ψε, QΩδVκΨε).
The left-hand side above is estimated as
(Ψε, QΩ(Heff − infσ(Hε(κ)))Ψε) ≥ (Σ + δ − infσ(Hε(κ)))(Ψε, QΩΨε).
117
Note that
Σ + δ − infσ(Hε(κ)) ≥ Σ + δ − Σ− 3
2θκ = δ − 3
2θκ > 0.
Then
(Ψε, QΩ(Heff − infσ(Hε(κ)) + g)Ψε) ≥ (δ − 3
2θκ)‖QΩΨε‖2 > 0.
Moreover
|(Ψε, QΩδVκΨε)| = |(δVκQΩΨε,Ψε)| ≤ ‖δVκQΩΨε‖‖Ψε‖
≤ θκ
(‖Hf
1/2QΩΨε‖+ ‖QΩΨε‖)‖Ψε‖
= θκ‖QΩΨε‖‖Ψε‖ ≤ θκ‖Ψε‖2.
Hence we have
0 < (δ − 3
2θκ)‖QΩΨε‖2 ≤ θκ‖Ψε‖2.
The lemma follows. 2
We normalize Ψε, i.e., ‖Ψε‖ = 1. Take a subsequence ε′ such that Ψε′ weakly
converges to a vector ϕg as ε′ →∞.
Proposition 4.21 [AH97, Lemma 4.9] Let Sn and S be self-adjoint operators on
a Hilbert space h, which have a common core D such that Sn → S on D strongly
as n → ∞. Let ψn be a normalized eigenvector of Sn such that Sn = Enψn, E =
limn→∞En and the weak limit ψ = w − limn→∞ ψn 6= 0 exist. Then Sψ = Eψ. In
particular if En is the ground state energy, then E is the ground state energy of S
and ψ is a ground state of S.
Proof: Since Sn converges to S in the strong resolvent sense by the assumption, we
can see that limn→∞(φ, (Sn−z)−1ψn) = (φ, (Sn−z)−1ψ) for any φ ∈ h. This implies
that (Sn − z)−1ψ = (E − z)−1ψ and then Sψ = Eψ. 2
Now we are in the position to state the main theorem in Section 4.
Theorem 4.22 (Enhanced binding) [HS01] Suppose Assumptions 3.4 and 4.9.
Then for any ε > 0, there exists κε such that for all κ > κε, H(κ) has a unique
ground state for all α such that |α| > αε.
118
Proof: Let Eε = infσ(Hε(κ)) and E = infσ(H(κ)). Since Hε(κ)→ H(κ) as ε→ 0 in
the strong resolvent sense, lim supε→0Eε ≤ E follows. On the other hand we notice
that Eε ≥ E + ε(Ψε, NΨε). Since ε(Ψε, NΨε) → 0 as ε → 0, lim infε→0Eε ≥ E
follows. Thus limε→0Eε = E. By Proposition 4.21 it is enough to prove ϕg 6= 0.
Note that N + PΩ ≥ 1l. Hence
1l⊗N + (EHeff
[Σ,Σ+δ) + EHeff
[Σ+δ,∞))⊗ PΩ = 1l⊗N + EHeff
[Σ,Σ+δ) ⊗ PΩ +QΩ ≥ 1l,
and
EHeff
[Σ,Σ+δ) ⊗ PΩ ≥ 1l− 1l⊗N −QΩ. (4.41)
Suppose that min|Σ|/3, 2κ2 > θκ. and δ > 32θκ. Then for ε′ such that |Σ| >
3θκ + ε′, we have by (4.41), Lemmas 4.19 and 4.20,
(Ψε′ , EHeff
[Σ,Σ+δ) ⊗ PΩΨε′)
≥ 1− (Ψε′ , NΨε′)− (Ψε′ , QΩΨε′)
≥ 1− 1
κ2C‖Q/ω5/2‖(max
µ‖∇µV ‖∞)− θκ
δ − 32θκ.
Note that supα ‖Q/ω5/2‖ < ∞ and limκ→∞
θκδ − 3
2θκ
= 0 uniformly with respect to α.
Hence for sufficiently large κ, (Ψε′ , EHeff
[Σ,Σ+δ)⊗PΩΨε′) > η follows uniformly in ε′ and
α with some η > 0. Take ε′ → 0 on both sides above. Since EHeff
[Σ,Σ+δ) ⊗ PΩ is a
finite rank operator, we see that EHeff
[Σ,Σ+δ) ⊗ PΩΨε′ → EHeff
[Σ,Σ+δ) ⊗ PΩϕg strongly and
(ϕg, EHeff
[Σ,Σ+δ)⊗PΩϕg) > η. In particular ϕg 6= 0. Then ϕg is a ground state of H(κ).
2
We can also show the existence of ground state for the Pauli-Fierz Hamiltonian
without scaling parameter.
Theorem 4.23 (Enhanced binding, no scaling) [HS01] Let κ = 1, i.e., the
Hamiltonian is not scaled. Suppose Assumption 3.4, and (1) and (2) of Assumption
4.9, and that
limα→∞
‖Q/ωn/2‖ = 0, n = 3, 4, 5. (4.42)
Then there exists α∗ > αε such that for all α with |α| > α∗, H has a ground state.
119
Figure 9: Spectrum of massless Hamiltonian
Proof: By (4.42) we can see that θκ → 0 and ‖Q/ω5/2‖ → 0 as α → ∞. Then
Lemma 4.18 holds for sufficiently large α with κ = 1. Then the massive ground
state Ψε exists. Furthermore we have
(Ψε′ , EHeff
[Σ,Σ+δ) ⊗ PΩΨε′) ≥ 1− C‖Q/ω5/2‖(maxµ‖∇µV ‖∞)− θ1
δ − 32θ1
,
where θ1 is θκ with κ = 1. Since lim|α|→∞ ‖Q/ω5/2‖ = 0 and lim|α|→∞θ1
δ− 32θ1
= 0, we
can conclude that ϕg 6= 0 for sufficiently large |α|. Then the corollary follows. 2
An example of (4.42) is given in Example 4.10. See Figure 9 for the spectrum of
massless Pauli-Fierz Hamiltonian.
4.4 Transition from unbinding to binding
In the previous sections we show the absence and the existence of ground state.
Combining these results we can construct examples of the Pauli-Fierz Hamiltonian
having transition from unbinding to binding according to the value of coupling
constant α.
Lemma 4.24 Suppose Assumptions 3.4 and 4.1. Then H(κ) has no ground state
for all κ > 0 and all α such that |α| < α0.
Proof: Define the unitary operator uκ by (uκf)(x) = kd/2f(x/κ). Then we infer
Vκ = κ−2uκV u−1κ , −∆ = κ−2uκ(−∆)u−1
κ and
‖|Vκ|1/2(−∆)−1|Vκ|1/2‖ = κ−2‖uκ|V |1/2u−1κ (−∆)−1uκ|V |1/2u−1
κ ‖ = ‖K0‖.
Then the lemma follows from Theorem 4.6. 2
Theorem 4.25 (Transition from unbinding to binding) [HSS11] Suppose As-
sumptions 3.4 , 4.1 and 4.9. Let arbitrary δ > 0 be given. Then there exists an
external potential V and constants α+ > α− such that
120
(1) 0 < α+ − α− < δ;
(2) H has a ground state for |α| > α+ but no ground state for |α| < α−.
Proof: For δ > 0 we take ε > 0 such that αε−α0 < δ. Take a sufficiently large κ, and
set V (x) = V (x/κ)/κ2. Define H by the Pauli-Fierz Hamiltonian with potential V .
Then H has a ground state for |α| > αε by Theorem 4.22, and H has no ground
state for |α| < α0 by Lemma 4.24. Set αε = α+ and α0 = α−. Then the proof is
completed. 2
4.5 Enhanced binding by UV cutoff
We can also consider the enhanced binding by UV cutoff. In Example 3.10 we give
the example of UV cutoff function:
ϕ(k) = 1l[λ,Λ](k). (4.43)
In this section we suppose that ϕ is (4.43) and the dimension d = 3. Thus meff =
m+ 83πα2(Λ− λ) and we have the corollary below.
Corollary 4.26 (Absence of ground state) Suppose Assumptions 3.4 and 4.1
and
Λ <8
3πα−2(mc −m) + λ. (4.44)
Then H has no ground state.
Proof: (4.44) implies that meff < mc. Then the corollary follows from Theorem 4.6.
2
We can also show the existence of ground state for sufficiently large Λ.
Corollary 4.27 (Enhanced binding) [Hir03] Suppose Assumption 3.4, and (1)
and (2) of Assumption 4.9. Then there exists Λ∗ such that H has a ground state for
Λ > Λ∗.
Proof: We notice that
meff(k) = m+ α2 8π
3(Λ− λ)
− α2
2
8π
3
(|k| log
∣∣∣∣(|k|+ Λ)(|k| − λ)
(|k|+ λ)(|k| − Λ)
∣∣∣∣− iπ1l[λ,Λ](|k|)√|k|).
121
Then we have
‖Q/ωn/2‖2 = α2
∫λ≤|k|≤Λ
1
meff(k)2ω(k)ndk
and
1l[λ,Λ](k)1
meff(k)2ω(k)n≤(
3
4π2α2
)21
ω(k)n+1.
Since the right and side above is integrable for n = 3, 4 and 5. The the Lebesgue
dominated convergence theorem yields that limΛ→∞ ‖Q/ωn/2‖ = 0. Hence in a
similar way to Theorem 4.23 we can prove the corollary. 2
122
Part III
The Nelson model
5 The Nelson Hamiltonian
5.1 The Nelson Hamiltonian
We begin with giving the definition of the Nelson Hamiltonian. Let F be the Boson
Fock space over L2(Rd). The creation operator and the annihilation operator satisfy
canonical commutation relations on Ffin:
[a(f), a∗(g)] = (f , g), [a(f), a(g)] = 0 = [a∗(f), a∗(g)]. (5.1)
For h ∈ L2(Rd), the field operator is defined by
φ(h) =1√2
∫ (a∗(k)h(−k) + a(k)h(k)
)dk. (5.2)
Let Hf be the free field Hamiltonian with the dispersion relation ω(k) = |k|.In order to study the binding of N particle system by the linear coupling with
the scalar quantum field, N particles assumed to be independent of each other,
and then there is no external potential linking two particles. Thus the N particle
Hamiltonian Hp is defined by the self-adjoint operator on L2(RdN) by
Hp =N∑j=1
(− 1
2mj
∆j + Vj
), (5.3)
where mj is the mass of the j-th particle and Vj = Vj(xj) external potential de-
pending only on xj. Hamiltonian Hp does not necessarily have ground states. For
example, with sufficiently shallow potential Vj’s, Hp has no ground states.
The state space of the N particle coupled to the scalar quantum field is
H = L2(RdN)⊗F . (5.4)
Let
H0 = Hp ⊗ 1l + 1l⊗Hf (5.5)
be the non-interacting Hamiltonian.
123
Definition 5.1 (Nelson Hamiltonian) The Nelson Hamiltonian H on H is de-
fined by
H = H0 +HI, (5.6)
where HI is given by
HI =N∑j=1
αj
∫ ⊕RdN
φj(xj)dx.
Here αj’s are real coupling constants, we identify H as H ∼=∫ ⊕RdN Fdx and φj(x),
x ∈ Rd, is given by
φj(x) =1√2
∫ (a∗(k)λj(−k)e−ikx + a(k)λj(k)eikx
)dk.
We will give assumptions on λj later. Since the semigroup e−tH is ergodic, the
uniqueness of the ground state of H follows.
The existence of ground states of the Nelson Hamiltonian has been investigated
in the last decade. [BFS98-a, BFS98-b] proved the existence of ground states un-
der some conditions. [Ger00, Spo99] remove the weak coupling condition, namely
they show the existence of ground states of the Nelson Hamiltonian for arbitrary
values of a coupling constant. [Sas05] shows the existence of a ground state with
general external potentials including the Coulomb potential, and [HHS05] shows the
existence of a ground state without cutoffs.
5.2 Enhanced binding
As is mentioned in the previous section, the existence of the ground state of the
Nelson Hamiltonian has been proven under some general conditions. One of fun-
damental assumption among them is that Hp has a ground state. In this note we
remove this condition.
If there is no interaction between particles, the j-th particle is governed only
by the potential Vj. In this case, if Vj’s are sufficiently shallow, external potential∑Nj=1 Vj can not trap these particles. But if these particles attractively interact with
each other by an effective potential derived from the scalar quantum field, particles
close up with each other and behave just like as one particle with heavy mass
124
Figure 10: Enhanced binding by effective potential
∑Nj=1 mj. We will see that effective potential is of the form
Veff(x) = −1
4
N∑i 6=j
αiαj
∫Rd
λi(−k)λj(k)
ω(k)e−ik·(xi−xj)dk. (5.7)
Effective potential Veff depends on the choice of cutoff function λj’s. A typical
example of Veff is a three dimensional N -body smeared Coulomb potential:
Veff(x1, ..., xN) = − 1
8π
N∑i 6=j
αiαj|xi − xj|
$(|xi − xj|),
where $(|x|) > 0 holds for a sufficiently small |x|. For this case it is determined by
signs of α1, ..., αN whether Veff is attractive or repulsive for sufficiently small |xi−xj|.We can see from (5.15) that an identical sign of coupling constants and
suppλi ∩ suppλj 6= ∅, i 6= j,
derive attractive effective potentials, and which enhances binding of the system. If
N is large enough, this one particle has sufficiently heavy mass and is bounded by
external potential∑N
j=1 Vj, and finally it is trapped. See Figure 10. Heuristically
we see that
H ∼ − 1
2∑N
j=1mj
∆ +N∑j=1
Vj =1∑N
j=1mj
(−1
2∆ + (
N∑j=1
mj)N∑j=1
Vj
). (5.8)
125
5.3 Weak coupling limit
In this section we see the relationship between an enhanced binding and a weak
coupling limit, which has been seen in the case of the Pauli-Fierz model in Section
3.9.1. In our model under consideration, it is seen that the enhanced binding is
derived from the effective potential Veff which is the sum of potentials between two
particles. Alternatively the effective potential can be derived from a weak coupling
limit [Dav77, Dav79, Hir98, Hir99], which is one of a key ingredient of this paper.
We outline a weak coupling limit by path measures. Let us introduce a scaling in
the Nelson model as
H(κ) = Hp + κ2Hf + κHI, (5.9)
where κ > 0 is a scaling parameter. Let
X+ = C([0,∞);RdN)
be the set of continuous paths valued in RdN . Then e−tH(κ) can be expressed by a
path measure as
(f ⊗ Ω, e−tH(κ)g ⊗ Ω)H =
∫X+×RdN
f(X0)g(Xt)e−∫ T0 V (Xs)dseWκdWxdx, (5.10)
where Xt = (X1t , ..., X
Nt ), Xj
t (w) = wj(t) ∈ Rd, w = (w1, ..., wN) ∈ X+, denotes
the point evaluation of w ∈X+, dWx the Wiener measure on X starting from x at
t = 0 and
V (Xs) =N∑j=1
Vj(Xjs ), (5.11)
Wκ =1
4
N∑i,j=1
αiαj
∫ T
0
ds
∫ T
0
dt
∫Rdλi(−k)λj(k)κ2e−κ
2|s−t|ω(k)e−ik·(Xis−X
jt )dk. (5.12)
Informally taking κ→∞ in (5.12), we see that only the diagonal part of∫ T
0ds∫ T
0dt
survives and the off diagonal part is dumped by the factor
κ2e−κ2|s−t|ω(k) → δ(s− t) 2
ω(k)
126
as κ→∞. Thus we have
Wκ →1
2
N∑i,j=1
αiαj
∫ T
0
ds
∫Rd
λi(−k)λj(k)
ω(k)e−ik·(X
is−X
js )dk (5.13)
as κ→∞. Combining the right-hand side of (5.13) with∫ T
0V (Xs)ds in (5.10), we
can derive the Feynman-Kac type formula:
limκ→∞
(5.10) =
∫X+×RdN
f(X0)g(Xt)e−∫ T0 (V (Xs)+Veff(Xs)+G)dsdWxdx, (5.14)
where
Veff(x1, ..., xn) = −1
4
N∑i 6=j
αiαj
∫Rd
λi(−k)λj(k)
ω(k)e−ik·(xi−xj)dk (5.15)
and G is the constant derived from the diagonal part of (5.13), which is given by
G = −1
4
N∑j=1
α2j
∫Rd
λj(−k)λj(k)
ω(k)dk. (5.16)
Note that when suppλi ∩ suppλj = ∅, i 6= j, the effective potential Veff vanishes and
only the constant G remains. Let
Heff =N∑j=1
(− 1
2mj
∆j + Vj
)+ Veff . (5.17)
Actually (5.14) can be shown rigorously.
Proposition 5.2 (Weak coupling limit) [Dav77, Dav79, Hir98, Hir99] Let t > 0.
Then
s-limκ→∞
e−tH(κ) = e−t(Heff+G) ⊗ PΩ,
where PΩ denotes the projection onto zΩ|z ∈ C ⊂ F . In particular for f, g ∈L2(RdN),
limκ→∞
(f ⊗ Ω, e−tH(κ)g ⊗ Ω)H = (f, e−t(Heff+G)g)L2(RdN ).
127
Proposition 5.2 is interesting in both the stochastic analysis and the operator
theory. Probabilistically, through a weak coupling limit as is seen in Proposition
5.2, one can derive a Markov process from a non Markov process. We will see it
below. The family of measures µxκκ>0 on the path space X+ is given by
µxκ(dX) = e−∫ t0 V (Xs)dseWκdWx. (5.18)
The double integral Wκ in (5.18) is independent of x and breaks a Markov property
of the stochastic process (Xs)s>0, and
Tκ,s : f 7−→∫
X+
f(Xs)µκ(dX)
does not define a semigroup on L2(RdN). By Proposition 5.2, however, the Markov
property revives as κ→∞, and we have T∞,s = e−s(Heff+G) .
Furthermore Proposition 5.2 also suggests that H(κ) ∼ Heff +G for a sufficiently
large κ. Actually we can show that H(κ) is isomorphic to a self-adjoint operator of
the form
Heff + κ2Hf +1
κH1 +
1
κ2H2 + constant (5.19)
with some operators H1 and H2. It is checked that under some condition Heff has
a ground state for αj’s such that 0 < αc < |αj|, j = 1, . . . , N , for some αc, which
suggests by (5.19) that for a sufficiently large κ, H(κ) also has a ground state for
αj with αc < |αj| < αc(κ), j = 1, ..., N , for some αc(κ). This is actually proved
by checking stability conditions for (5.19) under some assumptions. This is an idea
to show the enhanced binding for the Nelson model. Note that we do not need to
assume the existence of ground state of Hp, namely H(κ) with α1 = · · · = αN = 0
may have no ground state.
128
6 Binding
6.1 Existence of ground states
In order to show the enhanced binding we check the so-called stability condition.
The stability condition implies that the lowest two cluster threshold of H is strictly
larger than the ground state energy of H. Then intuitively atom can not be ionized
and thus the ground state is stable. We introduce assumptions:
Assumption 6.1 For all j = 1, ..., N , (i),(ii),(iii) and (iv) are fulfilled.
(i) λj(−k) = λj(k) and λj, λj/√ω ∈ L2(Rd).
(ii) There exists an open set S ⊂ Rd such that S = suppλj and λj ∈ C1(S).
(iii) For all R > 0, SR = k ∈ S||k| < R has a cone property.
(iv) For all p ∈ [1, 2) and all R > 0, |∇kλj| ∈ Lp(SR).
Condition (i) guarantees that HI is a symmetric operator. In order to show the
existence of a ground state, we applied a method invented in [GLL01]. Precisely,
we used the photon derivative bound and the Rellich-Kondrachov theorem. The
conditions (ii)-(iv) are required to verify these procedures in the proof of Proposition
6.4 below. It is easily proven that H is self-adjoint on D(H) = D(Hp)∩D(Hf) and
bounded from below for an arbitrary αj ∈ R.
Assumptions (V1) and (V2) are also introduced:
(V1) There exists αc > 0 such that inf σ(Heff) ∈ σdisc(Heff) for αj with |αj| > αc,
j = 1, ..., N .
(V2) Vj(−∆ + 1)−1, j = 1, . . . , N , are compact.
The main theorem in Section 6 is stated below.
Theorem 6.2 (Enhanced binding) [HS08] Let λj/ω ∈ L2(Rd), j = 1, ..., N , and
assume Assumption 6.1,(V1) and (V2). Fix a sufficiently large κ > 0. Then there
exists αc(κ) such that for αj with αc < |αj| < αc(κ), j = 1, ..., N , H(κ) has a ground
state, where αc(κ) is possibly infinity.
129
Proof: We give a proof in Section 6.2 2
The scaling parameter κ in Theorem 6.2 can be regarded as a dummy and ab-
sorbed into mj, Vj and λj, j = 1, ..., N . Let κ be sufficiently large. Define
H =N∑j=1
(− 1
2mj
∆j + Vj
)+
N∑j=1
αjφj +Hf ,
where mj = mjκ2, Vj = Vj/κ
2 and φj is defined by φj with λj replaced by λj/κ.
Corollary 6.3 Let λj/ω ∈ L2(Rd), j = 1, ..., N , and assume Assumption 6.1,(V1)
and (V2). Then H has a ground state for αc < |αj| < αc(κ), j = 1, ..., N , where
αc(κ) is introduced in Theorem 6.2.
Proof: We have κ−2H(κ) = H. Then by Theorem 6.2, H has a ground state. 2
6.2 Stability conditions
Let λj/ω ∈ L2(Rd), j = 1, ..., N , and define the unitary operator T on H by
T = exp
(−i1κ
N∑j=1
αjπj
),
where
πj =
∫ ⊕RdN
πj(xj)dx
with
πj(x) =i√2
∫ (a∗(k)e−ik·x
λj(−k)
ω(k)− a(k)eik·x
λj(k)
ω(k)
)dk.
Then we can show that T maps D(H) onto itself and
T−1H(κ)T =N∑j=1
1
2mj
(−i∇j −
αjκAj
)2
+ Vj −α2j
2‖λj/√ω‖2
+ κ2Hf + Veff
= Heff + κ2Hf +H ′(κ),
130
where Aj =
∫ ⊕RdN
Aj(xj)dx with
Aj(x) =1√2
∫k
(a∗(k)e−ikx
λj(−k)
ω(k)+ a(k)eikx
λj(k)
ω(k)
)dk
and
H ′(κ) =N∑j=1
1
κ
αj2mj
((−i∇j)·Aj + Aj ·(−i∇j)) +1
κ2
α2j
2mj
A2j −
α2j
2‖λj/√ω‖2
.
Let us set CN = 1, ..., N. For β ⊂ CN , we define
H0(β) = H0(β, κ) =∑j∈β
1
2mj
(−i∇j −
αjκAj
)2
+ κ2Hf + Veff(β),
Veff(β) =
−1
4
∑i,j∈β,i 6=j
αiαj
∫Rd
λi(−k)λj(k)
ω(k)e−ik·(xi−xj)dk, |β| ≥ 2,
0, |β| = 0, 1,
HV (β) = HV (β, κ) = H0(β) +∑j∈β
Vj,
where |β| = #β. Simply we set HV = HV (CN).
HV = H(κ)− 1
4
N∑j=1
α2j‖λj‖2 (6.1)
has ground states if and only if H(κ) does, since∑N
j=1 α2j‖λj‖2/4 is a fixed number.
The operators H0(β) and HV (β) are self-adjoint operators acting on L2(Rd|β|)⊗F .
We set
EV (κ) = infσ(HV ),
EV (κ, β) = inf σ(HV (β)),
E0(κ, β) = inf σ(H0(β)),
EV (κ, ∅) = 0.
The lowest two cluster threshold ΣV (κ) is defined by
ΣV (κ) = minEV (κ, β) + E0(κ, βc)|β $ CN. (6.2)
To establish the existence of ground state of H(κ), we use the next proposition:
131
Figure 11: Ionization EV (βc) + E(β)
Proposition 6.4 [GLL01] Let ΣV (κ)−EV (κ) > 0. Then H(κ) has a ground state.
For β ⊂ CN , we set the Schrodinger operators in L2(Rd|β|) by
h0(β) = −∑j∈β
1
2mj
∆j + Veff(β),
hV (β) = h0(β) +∑j∈β
Vj,
E0(β) = inf σ(h0(β)),
EV (β) = inf σ(hV (β)),
where h0(∅) = 0 and hV (∅) = 0. Furthermore we simply put
hV = hV (CN) = Heff , EV = inf σ(hV ). (6.3)
We define the lowest two cluster threshold for hV by (Figure 11)
ΞV = minEV (β) + E0(βc)|β $ CN (6.4)
and we set
Veff ij(x) = −1
4αiαj
∫Rd
λi(−k)λj(k)
ω(k)e−ik·xdk, i 6= j.
Lemma 6.5 Effective potentials Veff ij, i, j = 1, ..., N , are relatively compact with
respect to −∆.
132
Proof: Since λiλj/ω ∈ L1(Rd), i, j = 1, ..., N , we can see that Veff ij(x) is continuous
in x and lim|x|→∞ Veff ij(x) = 0 by the Riemann-Lebesgue theorem. In particular
Veff ij is relatively compact with respect to the d-dimensional Laplacian. 2
We want to estimate infσess(Heff). For Hamiltonians with the center of mass
motion removed, the bottom of the essential spectrum is estimated by HVZ theorem.
Lemma 6.6 Assume (V2). Then σess(Heff) = [ΞV ,∞).
Proof: We may assume that Vi, Veff ij ∈ C∞0 (Rd) by Proposition 6.17 below. Then
there exists a normalized sequence gnn ⊂ C∞0 (RdN) such that
suppgn ⊂x ∈ RdN
∣∣Vi(x) = 0, Veff ij(xi − xj) = 0, i, j = 1, ..., N
and (gn, hV (β)gn) =∑
j∈β
(gn,− 1
2mj∆gn
)→ 0 as n→∞. Then we have
EV (β) + E0(βc) ≤ 0. (6.5)
Let jβ ∈ C∞(Rd), β ∈ CN , be a Ruelle-Simon partition of unity, which satisfy (i)-(v)
below:
(i)∑β⊆CN
jβ(x)2 = 1,
(ii) jβ(Cx) = jβ(x) for |x| = 1, C ≥ 1 and β 6= CN ,
(iii) supp jβ ⊂x ∈ Rd
∣∣∣∣ mini∈β,j∈βc
|xi − xj|, |xj| ≥ c|x|
for some c > 0,
(iv) jβ(x) = 0 for |x| < 1
2and β 6= CN ,
(v) jCN has a compact support.
For a constant R > 0 we put jβ(x) = jβ(x/R). Note that for each β ⊂ CN ,
Heff = hV (β)⊗ 1l + 1l⊗ h0(βc) + Iβ,
where
Iβ =∑i∈βc
1l⊗ Vi(xi) +∑
i∈β,j∈βc
i∈βc,j∈β
Veff ij(xi − xj).
133
Here we identify as L2(RdN) ∼= L2(Rd|β|) ⊗ L2(Rd|βc|). By the IMS localization
formula [CFKS87, Theorem 3.2 and p. 34], we have
Heff = jCNHeffjCN +∑β$CN
jβ (hV (β)⊗ 1l + 1l⊗ h0(βc)) jβ+∑β$CN
j2βIβ−
1
2
∑βjCN
|∇jβ|2.
Since j2CN
(∑N
j=1 Vj+Veff) and∑
β$CN j2βIβ are relatively compact with respect to the
dN -dimensional Laplacian by the property (iii) and (v), it is seen that the essential
spectrum of Heff coincides with that of
jCN
(−1
2
N∑j=1
∆j
)jCN +
∑β$CN
jβ (hV (β)⊗ 1l + 1l⊗ h0(βc)) jβ −1
2
∑βjCN
|∇jβ|2.
We have ∑β$CN
jβ (hV (β)⊗ 1l + 1l⊗ h0(βc)) jβ ≥∑β$CN
(EV (β) + E0(βc))j2β.
By (ii) and (v), ∥∥∥1
2
∑βjCN
|∇jβ|2∥∥∥ ≤ C
R2
with some constant C independent of R. Hence we obtain that
inf σess(Heff) ≥ minx∈Rd
∑β$CN
(EV (β) + E0(βc))j2β −
C
R2≥ ΞV −
C
R2
for all R > 0. Here we used (i) and (6.5). Thus σess(Heff) ⊂ [ΞV ,∞) follows.
Next we shall prove the reverse inclusion σess(Heff) ⊃ [ΞV ,∞). Fix β $ CN . Let
ψVn ∞n=1 ⊂ C∞0 (Rd|β|) be a minimizing sequence of hV (β) so that
limn→∞
‖(hV (β)− EV (β))ψVn ‖ = 0, ‖ψVn ‖ = 1
and ψ0n∞n=1 ⊂ C∞0 (Rd|βc|) a normalized sequence such that
limn→∞
‖(h0(βc)− E0(βc)−K)ψ0n‖ = 0, (6.6)
where K ≥ 0 is a constant. Note that since σ(h0(βc)) = [E0(βc),∞), ψ0n such as
(6.6) exists. By the translation invariance of h0(βc), for any function τ· : N → Rd,
134
the shifted sequence ψ0n(xj1 − τn, . . . , xj|βc| − τn) also satisfies (6.6). Let Rn > 0 be
a constant satisfying
supp ψVn ⊂x = (xj1 , · · · , xj|β|) ∈ Rd|β| ||xji | < Rn, ji ∈ β, i = 1, ..., |β|
.
We take τ such that
supp ψ0n(· − τn, · · · , · − τn)
⊂x = (xk1 , · · · , xk|βc|) ∈ Rd|βc| ||xki | ≥ Rn + n, ki ∈ βc, i = 1, ..., |βc|
.
We set Ψn(x1 · · ·xN) = ψVn (xj1 · · ·xj|β|) ⊗ ψ0n(xk1 − τn · · ·xk|βc| − τn) ∈ L2(RdN).
Then, for all i, j with i ∈ β, j ∈ βc, we have
‖Veff ij(xi − xj)Ψn‖ ≤ supx∈Rd,|x|>n
|Veff ij(x)| → 0, (n→∞),
‖Vj(xj)Ψn‖ ≤ supx∈Rd,|x|≥Rn+n
|Vj(x)| → 0, (n→∞).
Hence, by a triangle inequality, we have that
‖(Heff − EV (β)− E0(βc)−K)Ψn‖ → 0, (n→∞).
Therefore [EV (β)+E0(βc)+K,∞) ⊂ σ(Heff). Since β $ CN and K > 0 are arbitrary,
[ΞV ,∞) ⊂ σess(Heff) follows. Thus the proof is complete. 2
We define ∆p(α1, ..., αN) = ΞV − EV .
Corollary 6.7 Assume (V1) and (V.2). Then ∆p(α1, ..., αN) > 0 follows for αjwith |αj| > αc, j = 1, ..., N .
Proof: Since infσess(Heff) = ΞV by Lemma 6.6 and infσ(Heff) ∈ σdisc(Heff) by (V1),
the corollary follows from ∆P (α1, ..., αN) = infσess(Heff)− infσ(Heff) > 0. 2
Lemma 6.8 For an arbitrary κ > 0, it follows that ΣV (κ) ≥ ΞV .
Proof: It is well known that HV (β) can be realized as a self-adjoint operator on a
Hilbert space HQ = L2(R|β|d)⊗L2(Q, dµ) with some probability space (Q, µ), which
is called a Schrodinger representation. It is established in e.g., [LHB11] that
(Ψ, e−tHV (β)Φ)HQ≤ (|Ψ|, e−t(hV (β)+κ2Hf)|Φ|)HQ
.
135
Hence for any β ⊂ CN , it follows that
infσ(hV (β) + κ2Hf) ≤ infσ(HV (β)).
Since infσ(Hf) = 0 and infσ(hV (β)⊗ 1l + κ21l⊗Hf) = infσ(hV (β)), we can obtain
infσ(hV (β)) ≤ infσ(HV (β))
for arbitrary β ∈ CN . Then the lemma follows from the definition of lowest two
cluster thresholds. 2
Lemma 6.9 Assume (V1). Then
E(κ) ≤ EV + κ−2
N∑j=1
α2j‖λj‖2/(4mj)
for αj with |αj| > αc, j = 1, ..., N .
Proof: By (V1), Heff has a normalized ground state u for αj with |αj| > αc, j =
1, ..., N . Set Ψ = u⊗ Ω. Then
E(κ) ≤ (Ψ, H(κ)Ψ) ≤ (u,Heffu) +N∑j=1
αj2mjκ
2<(i∇jΨ, AjΨ) +N∑j=1
α2j
2mjκ2‖AjΨ‖2
= EV +N∑j=1
α2j
4mjκ2‖λj‖2.
Here we used that (∇jΨ, AjΨ) =1√2
d∑µ=1
(∇xjµu ⊗ Ω, u ⊗ a∗(kµe
−ik·xλj/ω)Ω) = 0.
Then the lemma follows. 2
Proof of Theorem 6.2
By Lemmas 6.8 and 6.9, we have
ΣV (κ)− E(κ) ≥ ΞV − EV −N∑j=1
α2j
4mjκ2‖λj‖2 = ∆p(α1, ..., αN)−
N∑j=1
α2j
4mjκ2‖λj‖2.
Note that ∆p(α1, ..., αN) > 0 is continuous in α1, ..., αN . Then for a sufficiently
large κ, there exists αc(κ) > αc such that for αc < |αj| < αc(κ), j = 1, ..., N ,
ΣV (κ)−E(κ) > 0. Thus H(κ) has a ground state for such αj’s by Proposition 6.4.
2
136
6.3 Examples
6.3.1 Example of effective potential
We show a typical example of cutoff function and effective potentials. We intro-
duce the assumption below. Let λj = ρj/√ω, j = 1, ..., N , with rotation invariant
nonnegative functions ρj. In this case, by (5.15), effective potential Veff is explicitly
computed as
Veff(x1, · · · , xN)
= −1
4
N∑i 6=j
αiαj
√(2π)d
|xi − xj|(d−1)/2
∫ ∞0
r(d−1)/2
r2ρi(r)ρj(r)
√r|xi − xj|J d−2
2(r|x|)dr.
(6.7)
Here Jν is the Bessel function: Jν(x) = (x2)ν∑∞
n=0(−1)n
n!Γ(n+ν+1)(x
2)2n. We can see that
Veff satisfies that
(1) Veff ij is continuous,
(2) lim|x|→∞ Vij(x) = 0,
(3) Veff ij(0) < Veff ij(x) for all x ∈ Rd but x 6= 0.
In particular, when d = 3 and ρj is the indicator function such as
ρj(k) =
0 |k| < κ,
1/√
(2π)3 κ < |k| < Λ,0 |k| ≥ Λ,
(6.8)
we see that
Veff(x1, · · · , xN) = − 1
8π2
N∑i 6=j
αiαj|xi − xj|
∫ Λ|xi−xj |
κ|xi−xj |
sin r
rdr. (6.9)
For sufficiently small |xi − xj|, i, j = 1, ..., N , and αj with an identical sign, the
effective potential (6.9) is attractive.
137
6.3.2 Example of external potential
We give an example of V1, · · · , VN satisfying assumption (V1). Assume simply that
V1 = · · · = VN = V , α1 = · · · = αN = α, λ1 = · · · = λN = λ and m1 = · · · = mN =
m. Then
Veff ij(x) = W (x) = −α2
4
∫Rd
|λ(k)|2
ω(k)e−ik·xdk
for all i 6= j. Let
hV (α) =N∑j=1
(− 1
2m∆j + V (xj)
)+ α2
N∑j 6=l
W (xj − xl),
which acts on L2(RdN). We assume (W1)-(W3) below:
(W1) V is relatively compact with respect to the d-dimensional Laplacian ∆, and
σ(−(∆/2m) + V ) = [0,∞).
(W2) W satisfies that −∞ < W (0) = ess. inf|x|<ε
W (x) < ess. inf|x|>ε
W (x) for all ε > 0.
(W3) infσ(−(∆/(2Nm) +NV ) ∈ σdisc(−(∆/(2Nm) +NV ).
Remark 6.10 Note that examples of Veff given in (6.7) satisfies (W2), and remem-
ber that lim|x|→∞W (x) = 0 and W (x) is relatively compact with respect to the
d-dimensional Laplacian. See Lemma 6.5. The condition (W1) means that the
external potential V is shallow and the non-interacting Hamiltonian hV (0) has no
negative energy bound state.
Theorem 6.11 Assume (W1)-(W3). Then, there exists αc > 0 such that for all α
with |α| > αc, infσ(hV (α)) ∈ σdisc(hV (α)). Namely hV (α) for |α| > αc has a ground
state.
To prove Theorem 6.11 we need several lemmas. For β ⊂ CN , we define
h0(α, β) = − 1
2m
∑j∈β
∆j + α2∑j,l∈βj 6=l
W (xj − xl),
hV (α, β) = h0(α, β) +∑j∈β
V (xj),
E0(α, β) = inf σ(h0(α, β)),
EV (α, β) = inf σ(hV (α, β)),
138
where EV (α, ∅) = 0 and E0(α, ∅) = 0. Simply we set EV (α,CN) = EV (α) and
E0(α,CN) = E0(α). Let ΞV (α) denote the lowest two cluster threshold of hV (α)
defined by (6.4). Then by (W1) and Lemma 6.6, we have
σess(hV (α)) = [ΞV (α),∞). (6.10)
Lemma 6.12 Let β $ CN but β 6= ∅. Then there exists a1 > 0 such that, for all α
with |α| > a1,
E0(α) < EV (α, β) + E0(α, βc). (6.11)
Proof: Since h0(α, β)/α2 and hV (α, β)/α2 converge to∑
j,l∈βj 6=l
W (xj − xl) in the uni-
form resolvent sense, by (W2), one can show that
limα→∞
EV (α, β)
α2= lim
α→∞
E0(α, β)
α2= |β|(|β| − 1)W (0).
2
Hence
limα→∞
E0(α)
α2= N(N − 1)W (0)
and
limα→∞
1
α2(EV (α, β) + E0(α, βc)) =
(|β|(|β| − 1) + |βc|(|βc| − 1)
W (0)
=N(N − 1) + 2|β|(|β| −N)
W (0).
Since |β|(|β| − N) ≤ −1 and W (0) < 0 by (W2), we see that there exists a1 > 0
such that (6.11) holds for all α with |α| > a1. 2
Let X = (x1, ..., xN)t ∈ RdN and Y = (xc, y1, . . . , yN−1)t be its Jacobi coordi-
nates:
xc =1
N
N∑j=1
xj, yj = xj+1 −1
j
j∑i=1
xi, j = 1, ..., N − 1.
139
Let T ∈ GL(N,R) be such that Y = TX. Note that
T =
1N
1N
1N
· · · · · · · · · 1N
−1 1 0 · · · · · · · · · 0−1
2−1
21 0 0 · · · 0
−13
−13
−13
1 0 · · · 0...
...... · · · . . . · · · · · ·
......
... · · · · · · . . . · · ·− 1N−1
− 1N−1
− 1N−1
· · · · · · − 1N−1
1
and
T−1 =
1 −12−1
3−1
4−1
5· · · · · · − 1
N
1 12−1
3−1
4−1
5· · · · · · − 1
N
1 0 23−1
4−1
5· · · · · · − 1
N
1 0 0 34−1
5−1
6· · · − 1
N...
... · · · · · · . . . · · · · · · ......
... · · · · · · · · · . . . · · · ...1 0 · · · · · · · · · 0 N−2
N−1− 1N
1 0 · · · · · · · · · · · · 0 N−1N
.
Matrix T induces the unitary operator U : L2(RdNX )→ L2(RdN
Y ) defined by
(Uψ)(Y ) = ψ T−1(Y ).
We have
Uh0(α)U−1 = − 1
2Nm∆xc −
N∑j=1
1
2µj∆yj + α2
N∑j 6=l
W (xj(Y )− xl(Y )),
UhV (α)U−1 = Uh0(α)U−1 +N∑j=1
V (xj(Y )),
where µj = jm/(j + 1) is a reduced mass and xj(Y ) = (T−1Y )j. Let k(α) be h0(α)
with the center of mass motion removed:
k(α) = −N∑j=1
1
2µj∆yj + α2
N∑j 6=l
W (xj(Y )− xl(Y )).
140
Set RdN = Rdxc ⊕ Rd(N−1)
y1,...,yN−1 = χc ⊕ χ⊥c . Since xj(Y ) − xi(Y ), i, j = 1, ..., N − 1,
depend only on y1, . . . , yN−1 ∈ χ⊥c , k(α) is a self-adjoint operator acting in L2(χ⊥c ).
Lemma 6.13 There exists a2 > 0 such that infσ(k(α)) ∈ σdisc(k(α)) for all α with
|α| > a2.
Proof: Note that lim|x|→∞W (x) = 0. Let χ, χ ∈ C∞(R) be such that χ(x)2+χ(x)2 =
1 with χ(x) =
1, |x| < 1,
0, |x| > 2.For a parameter R, we set
χR(y1) = χ(|y1|/R), χR(y1) = χ(|y1|/R), y1 ∈ Rd,
θR(Y1) = χ(|Y1|/2R), θR(Y1) = χ(|Y1|/2R), Y1 = (y2, . . . , yN−1) ∈ Rd(N−2).
By the IMS localization formula, we have
k(α) = χRθRk(α)θRχR + χRθRk(α)θRχR + χRk(α)χR +B(R), (6.12)
where
B(R) = −1
2χ2R|∇θR|2 −
1
2χ2R|∇θR|2 −
1
2|∇χR|2 −
1
2|∇χR|2.
Here B(R) : L2(χ⊥c )→ L2(χ⊥c ) is a bounded operator with the bound
‖B(R)‖ ≤ C
R2,
where C is a constant independent of R. Let us define k′(α) by k(α) with the firs
term χRθRk(α)θRχR in (6.12) replaced by χRθR(−∑N
j=1(1/2µj)∆yj)θRχR;
k′(α) = χRθR
(−
N∑j=1
1
2µj∆yj
)θRχR + χRθRk(α)θRχR + χRk(α)χR +B(R).
Since the difference between k(α) and k′(α) is χ2Rθ
2Rα
2∑N
j 6=lW (xj(Y )− xl(Y )), and
which is relatively compact with respect to the kinetic term −∑N
j=1(2µj)−1∆yj by
Remark 6.10, we have σess(k(α)) = σess(k′(α)). Moreover k′(α) can be estimated
141
from below as
k′(α) ≥χ2Rθ
2RE(k(α)− α2W (x2(Y )− x3(Y ))− α2W (x3(Y )− x2(Y ))
)(6.13)
+ χ2Rθ
2Rα
2(W (x2(Y )− x3(Y )) +W (x3(Y )− x2(Y ))
)(6.14)
+ χ2RE(k(α)− α2W (x1(Y )− x2(Y ))− α2W (x2(Y )− x1(Y ))
)(6.15)
+ χ2Rα
2(W (x1(Y )− x2(Y )) +W (x2(Y )− x1(Y ))
)(6.16)
− C/R2. (6.17)
Note that y1 = x2(Y )− x1(Y ) and x3(Y )− x2(Y ) = y2 − y1/2. We have
|(6.14)| ≤ 2 supy1,y2
|y1|<2R, |y2|>4R
α2|W (y2 − y1/2)| ≤ 2α2 sup|y|>3R
|W (y)|,
|(6.16)| ≤ 2 sup|y1|>2R
α2|W (y1)|.
Since we assume that lim|x|→∞W (x) = 0, we obtain that limR→∞ ‖(6.14)‖ = 0 and
limR→∞ ‖(6.16)‖ = 0. Thus, for all R > 0 we have
inf σess(k(α)) = inf σess(k′(α))
≥ infY ∈Rd(N−1)
[(6.13) + (6.15)]− ‖(6.14)‖ − ‖(6.16)‖ − C/R2
≥ minE(k(α)− α2W (x1 − x2)− α2W (x2 − x1)),
E(k(α)− α2W (x2 − x3)− α2W (x3 − x2))+ o(R), (6.18)
where limR→∞ o(R)/R = 0. It is seen that
limα→∞
1
α2E(k(α)− α2W (x1 − x2)− α2W (x2 − x1)
)= [N(N − 1)− 2]W (0), (6.19)
limα→∞
1
α2E(k(α)− α2W (x2 − x3)− α2W (x3 − x2)
)= [N(N − 1)− 2]W (0),
(6.20)
limα→∞
E(k(α))
α2= N(N − 1)W (0). (6.21)
Therefore combining (6.18)-(6.21) we obtain that
limα→∞
1
α2
(inf σess(k(α))− infσ(k(α))
)≥ −2W (0). (6.22)
142
Since W (0) < 0 by (W2), there exists a2 > 0 such that inf σess(k(α))−infσ(k(α)) > 0
for |α| > a2. This implies the desired result. 2
Lemma 6.14 Let uα be a normalized ground state of k(α), where |α| > a2. Then
|uα(y1, . . . , yN−1)|2 → δ(y1) · · · δ(yN−1) as α→∞ in the sense of distributions.
Proof: It suffices to show that for all ε > 0,
limα→∞
∫|Y0|>ε
|uα(Y0)|2dY0 = 0, Y0 = (y1, . . . , yN−1). (6.23)
We prove (6.23) by a reductive absurdity. Assume that
lim inf`→∞
∫|Y0|>ε
|uα`(Y0)|2dY0 > 0
for some constant ε > 0 and some sequence α`∞`=1 ⊂ R such that α` →∞(`→∞).
We can take a subsequence α`∞`=1 ⊂ α`∞`=1 so that
γ = lim`→∞
∫|Y0|>ε
|uα`(Y0)|2dY0 > 0.
Since k(α)/α2 ≥ N(N−1)W (0) and limα→∞E(k(α)/α2) = N(N−1)W (0), we have
N(N − 1)W (0) = lim`→∞
1
α2`
(uα` , k(α`)uα`) = lim`→∞
(uα` ,
N∑j 6=l
W (xj(Y0)− xl(Y0))uα`
)
≥ (1− γ)N(N − 1)W (0) + γ inf|Y0|>ε
N∑j 6=l
W (xj(Y0)− xl(Y0))
≥ N(N − 1)W (0).
Thus we have
inf|Y0|>ε
N∑j 6=l
W (xj(Y0)− xl(Y0)) = N(N − 1)W (0). (6.24)
By (W2) and (6.24) there exists a sequence Zn = (z1,n, . . . , z(N−1),n) ∈ Rd(N−1) such
that |Zn| > ε and limn→∞(xj(Zn) − xl(Zn)) → 0 for j 6= l. By the definition of
143
xj(Y ), we have
limn→∞
(x2(Zn)− x1(Zn)) = limn→∞
z1,n = 0,
limn→∞
(x3(Zn)− x2(Zn)) = limn→∞
(z2,n −1
2z1,n) = lim
n→∞z2,n = 0,
· · ·limn→∞
(xN(Zn)− xN−1(Zn)) = limn→∞
zN−1,n = 0.
This is a contradiction to |Zn| > ε > 0 for all n. 2
Proof of Theorem 6.11
Let uα be a ground state of k(α) = Uh0(α)U−1. By Proposition 6.17, we may
assume that V ∈ C∞0 (Rd). Let |α| > a2. Let v ∈ C∞0 (Rd) be a normalized vector
such that (v,
(− 1
2Nm∆xc +NV (xc)
)v
)< 0. (6.25)
Such a vector exists by (W3). We set Ψ(Y ) = Ψ(xc, Y0) = v(xc)uα(Y0) for Y =
(xc, Y0) = (xc, y1, . . . , yN−1) ∈ RdN . Then
(Ψ, UhV (α)U−1Ψ) = − 1
2mN(v,∆xcv) + E0(α) +
(Ψ,
N∑j=1
V (xj(Y ))Ψ
). (6.26)
We define
V αj,∗(xc) =
∫Rd(N−1)
dy1 · · · dyN−1V (xj(Y ))|uα(y1, . . . , yN−1)|2, j = 1, . . . , N.
By Lemma 6.14, we have
limα→∞
(Ψ,
N∑j=1
V (xj(Y ))Ψ
)= lim
α→∞
N∑j=1
(v, V αj,∗v) = (v,NV (xc)v).
Therefore, by (6.25) and (6.26), (Ψ, hV (α)Ψ) < E0(α) for |α| > a3 with some a3 > 0.
By this inequality, Lemma 6.12 and (6.10), we conclude that for α with |α| > αc =
maxa1, a3,ΞV (α)− EV (α) ≥ E0(α)− EV (α) > 0.
Then the theorem follows. 2
We give a general lemma.
144
Lemma 6.15 Let Kε, ε > 0, and K be self-adjoint operators on a Hilbert space
K and σess(Kε) = [ξε,∞). Suppose that limε→0Kε = K in the uniform resolvent
sense, and limε→0 ξε = ξ. Then σess(K) = [ξ,∞). In particular limε→0 infσess(Kε) =
infσess(K).
Proof: Let a > ξ. Then there exists ε0 such that for all ε with ε < ε0, ξε < a, from
which we have a ∈ σ(Kε) for all ε < ε0. Since Kε uniformly converges to K in the
resolvent sense, a ∈ σ(K) follows from [RS80, Theorem VIII.23 and p.291]. Since a
is arbitrary, (ξ,∞) ⊂ σ(K) follows and then
[ξ,∞) ⊂ σess(K).
It is enough to show infσess(K) = ξ. Let λ ∈ [infσess(K), ξ) but λ 6∈ σ(K). Note that
for all sufficiently small ε, λ 6∈ σ(Kε) by [RS80, Theorem VIII.24]. Since R \ σ(K)
is an open set, there exists δ > 0 such that (λ − δ, λ + δ) 6⊂ σ(K). Let PA(T )
denote the spectral projection of a self-adjoint operator T on a Borel set A ⊂ R. We
have limε→0 P(infσess(K)−δ′,λ)(Kε) = P(infσess(K)−δ′,λ)(K) uniformly by [RS80, Theorem
VIII.23 (b)]. In particular, for some δ′ > 0,
‖P(infσess(K)−δ′,λ)(Kε)− P(infσess(K)−δ′,λ)(K)‖ < 1.
Then P(infσess(K)−δ′,λ)(Kε)K is isomorphic to P(infσess(K)−δ′,λ)(K)K. Hence the dimen-
sion of P(infσess(K)−δ′,λ)(K)K is finite, since that of P(infσess(Kε)−δ′,λ)(K)K is finite.
Thus (infσess(K) − δ′, λ) ∩ σ(K) ⊂ σdisc(K). This is a contradiction. Hence we
have [infσess(K), ξ) ⊂ σ(K). Suppose that infσess(K) < ξ. Let τ > 0 be sufficiently
small. Note that (infσess(K)− τ, infσess(K) + τ) ⊂ σdisc(Kε) for all sufficiently small
ε. Let θ ∈ C∞0 (R) satisfy that
θ(z) =
1, |z − infσess(K)| < τ,0, |z − infσess(K)| > 2τ.
Then we have limε→0 θ(Kε) = θ(K) uniformly by [RS80, Theorem VIII.20]. Since
θ(Kε) is a finite rank operator for all sufficiently small ε, θ(K) has to be a com-
pact operator. It contradicts with the fact, however, that the spectrum of θ(K) is
continuous. Then we can conclude that infσess(K) = ξ and the proof is complete.
2
Let V : Rd → R be a real-valued measurable function.
145
Lemma 6.16 Let ∆ be the d-dimensional Laplacian. Assume that V (−∆ + 1)−1 is
a compact operator. Then there exists a sequence V εε>0 such that V ε ∈ C∞0 (Rd)
and limε→0 Vε(−∆ + 1)−1 = V (−∆ + 1)−1 uniformly.
Proof: Generally, let A be a compact operator and Bnn bounded operators such
that s-limn→∞Bn = 0, then BnA → 0 as n → ∞ in the operator norm. Since
V (−∆ + 1)−1 is a compact operator, we obtain that for a sufficiently large R > 0,
‖(1− χR)V (−∆ + 1)−1‖ < ε/3, (6.27)
where χR characteristic function of x ∈ Rd||x| < R. Let χ(n) denote the charac-
teristic function of x ∈ Rd||V (x)| < n. Since (1− χ(n))→ 0 strongly as n→∞,
‖(1− χ(n))χRV (−∆ + 1)−1‖ < ε/3 (6.28)
for a sufficiently large n. Since C∞0 (supp(χRχ(n))) is dense in L2(supp(χRχ
(n))),
there exists a sequence Vmm ⊂ C∞0 (supp(χRχ(n)))
‖Vm − χRχ(n)V ‖L2(Rd) → 0
as m→∞. Since χRχ(n)V has a compact support and is bounded, we obtain that
s-limm→∞ Vm = χRχ(n)V as an operator. Thus for a sufficiently large m,
‖(Vm − χRχ(n)V )(−∆ + 1)−1‖ < ε/3. (6.29)
By (6.27)-(6.29) we can obtain that for an arbitrary ε > 0, ‖(V −Vm)(−∆+1)−1‖ < ε
for a sufficiently large m. Thus the lemma follows by setting Vm = V ε. 2
Let β ⊂ CN . Set
k0(β) = −∑j∈β
1
2mj
∆j +∑i,j∈β
Vij, kV (β) = h0(β) +∑j∈β
Vj
with Vi ∈ L2loc(Rd) and Vij ∈ L2
loc(Rd) such that Vi(−∆ + 1)−1 and Vij(−∆ + 1)−1
are compact operators. We define K = kV (CN). Let
ΞV = minβ$CN
infσ(k0(β)) + infσ(kV (β)) (6.30)
be the lowest two cluster threshold of K.
146
Proposition 6.17 There exist sequences V εi ε, V ε
ijε ⊂ C∞0 (Rd), i, j = 1, ..., N ,
such that
(1) limε→0
ΞV (ε) = ΞV , (2) limε→0
infσess(K(ε)) = infσess(K),
where ΞV (ε) (resp. K(ε) ) is ΞV (resp. K) with Vi and Vij replaced by V εi and V ε
ij,
respectively.
Proof: By Lemma 6.16, there exist sequences V εi ε>0, V ε
ijε>0 ⊂ C∞0 (Rd), such that
V εi (xi)(−∆i + 1)−1 → Vi(xi)(−∆i + 1)−1
and
V εij(xi − xj)(−∆i −∆j + 1)−1 → Vij(xi − xj)(−∆i −∆j + 1)−1
uniformly as ε→ 0 for i, j = 1, ..., N . Hence infσ(kV (ε)) and infσ(k0(ε)) converge to
infσ(kV ) and infσ(k0) as ε → 0, respectively. Then (1) follows from the definition
(6.30). By this and the uniform convergence of K(ε) to K in the resolvent sense,
Lemma 6.15 yields (2). 2
147
148
7 Absence of ground state
7.1 Introduction
7.1.1 Stability and the decay of variable mass
In this section we study a general version of the Nelson model, i.e., the Nelson model
with a variable coefficients. This model is an extension of the standard Nelson model
when the Minkowskian space-time is replaced by a static pseudo Riemannian mani-
fold. It is studied in the series of papers [GHPS09, GHPS11, GHPS12-a, GHPS12-b].
In this section the absence of ground state of the Nelson model with variable coeffi-
cients is discussed under infrared singularity condition. Throughout this section we
assume that
d = 3.
The Hamiltonian of the Nelson Hamiltonian with a variable coefficients is defined
formally by
H =1
2
3∑µ,ν=1
DµAµν(x)Dν +W (x) +
∫ω(D, x)a∗(x)a(x)dx
+1√2
∫ω−1/2(D, x)ρ(x− x)(a∗(x) + a(x))dx, (7.1)
where a(x) and a∗(x) the annihilation operator and the creation operator in the
position representation, respectively, ρ a nonnegative cutoff function and ω = h1/2 a
dispersion relation with a position dependent variable mass m(x):
h = h(D, x) =3∑
µ,ν=1
c(x)−1Dµaµν(x)Dνc(x)−1 +m2(x). (7.2)
We give examples of (7.2) in the next section. In [GHPS11] the existence of ground
states of H is shown when
m(x) ≥ a〈x〉−1, (7.3)
where 〈x〉 = (1 + |x|2)1/2. Then we study the case of
m(x) ≤ a〈x〉−β/2, β < 2
in this lecture note. See Figure 12.
149
m(x) ≥ a〈x〉−1 m(x) ≤ a〈x〉−β, β > 1
ground state exist not exist
Figure 12: Existence and absence of ground state
The standard Nelson model is defined by H with Aµν(x) replaced by δµν , aµν(x)
by δµν and m(x) by a constant m ≥ 0. By the condition ρ ≥ 0, ρ(0) > 0 follows, and
the integral∫|ρ(k)|2/ω(k)3dk is finite if and only if m > 0 since d = 3. Thus m > 0
corresponds to the infrared regular condition and m = 0 to the infrared singular
condition.
7.1.2 Klein-Gordon equation on pseudo Riemannian manifold
In quantum field theory the dispersion relation ω =√−∆ +m2 can be derived from
the Klein-Gordon equation:∂2φ
∂t2= −ω2φ. (7.4)
On the other hand the dispersion relation with variable coefficients can be derived
from the Klein-Gordon equation on a pseudo Riemannian manifold. We here give an
example of a Klein-Gordon equation defined on a static pseudo Riemannian manifold
M such that a short range potential v(x) = O(〈x〉−β−2) appears.
Let x = (t, x) = (x0, x) ∈ R×R3 and M the 4 dimensional pseudo Riemannian
manifold equipped with the metric tensor:
g(x) = g(x) =
e−θ(x) 0 0 0
0 −e−θ(x) 0 00 0 −e−θ(x) 00 0 0 −e−θ(x)
. (7.5)
Note that g depends on x but independent of t. The line element associated with g
is given by
ds2 = e−θ(x)dt⊗ dt− e−θ(x)
3∑j=1
dxj ⊗ dxj. (7.6)
150
The Klein-Gordon equation on M is
gφ+m2φ = 0, (7.7)
where the d’Alembertian operator is defined by
g = eθ(x)∂2t − e2θ(x)
∑j
∂je−θ(x)∂j. (7.8)
Thus the Klein-Gordon equation (7.7) is reduced to the equation
∂2φ
∂t2= K0φ, (7.9)
where
K0 = eθ(x)∑j
∂je−θ(x)∂j − e−θ(x)m2. (7.10)
The operator K0 is symmetric on the weighted L2 space L2(Rd; e−θ(x)dx). Now we
transform the operator K0 to the one on L2(R3). This is done by the unitary map
U0 : L2(Rd; e−θ(x)dx)→ L2(Rd), f 7→ e−(1/2)θf .
Lemma 7.1 There exist functions θ and v such that U0K0U−10 = ∆ − v, v(x) =
O(〈x〉−β−2) for β ≥ 0, and −∆ + v has no non-positive eigenvalues.
Hence the Klein-Gordon equation (7.9) is transformed to the equation
∂2φ
∂t2−∆φ+ vφ = 0 (7.11)
on L2(R3), and the dispersion relation is given by√−∆ + v. Although the proof of
Lemma 7.1 is straightforward, we shall show this statement through a more general
scheme in what follows.
Suppose that g = (gµν), µ, ν = 0, 1, 2, 3, is a metric tensor on R4 such that
(1) gµν(x) = gµν(x), i.e., it is independent of time t,
(2) g0j(x) = gj0(x) = 0, j = 1, 2, 3,
(3) gij(x) = −γij(x), where γ = (γij) denotes a 3-dimensional Riemannian metric.
151
Namely
g =
[g00 00 −γ
]. (7.12)
Let M be a pseudo Riemannian manifold equipped with the metric tensor g satis-
fying (1)-(3) above. Then the line element on M is given by
ds2 = g00(x)dt⊗ dt−3∑
i,j=1
γij(x)dxi ⊗ dxj.
Let g−1 = (gµν) denote the inverse of g. In particular 1/g00 = g00. We also denote
the inverse of γ by γ−1 = (γij). The Klein-Gordon equation on the static pseudo
Riemannian manifold M is generally given by
gφ+ (m2 + ηR)φ = 0, (7.13)
where η is a constant, R the scalar curvature of M , and g is the d’Alembertian
operator on M , which is given by
g =3∑
µ,ν=0
1√|detg|
∂µgµν√|detg|∂ν . (7.14)
Let us assume that g00(x) > 0. Then (7.13) is rewritten as
∂2φ
∂t2= Kφ, (7.15)
where
K = g00
(1√|detg|
3∑i,j=1
∂j√|detg|γji∂i −m2 − ηR
). (7.16)
The operator K is symmetric on L2(R3; ρ(x)dx), where
ρ =
√|detg|g00
= g−1/200
√|detγ|. (7.17)
Now let us transform the operator K on L2(R3; ρ(x)dx) to the one on L2(R3). Define
the unitary operator U : L2(R3; ρ(x)dx)→ L2(R3) by
Uf = ρ1/2f. (7.18)
152
Let ρi = ∂iρ and ∂i∂jρ = ρij for notational simplicity. Furthermore we set αij =
g00γij and ∂kα
ij = αijk . Since U−1∂jU = ∂j +ρj2ρ
, we see that as an operator identity
U−1
(3∑
i,j=1
∂ig00γij∂j
)U = g00
3∑i,j=1
γij∂i∂j + V1 + V2, (7.19)
where
V1 =3∑
i,j=1
(αiji + αij
ρiρ
)∂j,
V2 =1
4
3∑i,j=1
(2αiji
ρjρ
+ 2αijρijρ− αij ρi
ρ
ρjρ
).
Set |detg| = G and ∂iG = Gi. Hence we have
V1 = g00
3∑i,j=1
(γiji +
Gi
2G
)∂j,
where γiji = ∂iγij, and directly we can see that
g001√|detg|
3∑i,j=1
∂i√|detg|γij∂j = V1 + g00
3∑i,j=1
γij∂i∂j. (7.20)
Comparing (7.19) with (7.20) we obtain that
U−1
(3∑
i,j=1
∂ig00γij∂j − V2
)U = g00
1√|detg|
3∑i,j=1
∂i√|detg|γij∂j. (7.21)
Then we proved the lemma below.
Lemma 7.2 It follows that
UKU−1 =3∑
i,j=1
∂ig00γij∂j − v, (7.22)
where v = g00(m2 + ηR) + V2.
153
By Lemma 7.2, (7.15) is transformed to the equation:
∂2φ
∂t2=
(3∑
i,j=1
∂ig00γij∂j − v
)φ (7.23)
on L2(Rd).
Proof of Lemma 7.1: Now we come back to the proof of Lemma 7.1. Set
gµν(x) =
e−θ(x), µ = ν = 0,−e−θ(x), µ = ν = 1, 2, 3,0, µ 6= ν.
Then
ρ =
√|detg|g00
= e−θ, αij = g00γij = δij, (7.24)
and UKU−1 = ∆− v follows by (7.22), where, inserting (7.24) to v, we have
v = e−θ(m2 + ηR)− ∆θ
2+|∇θ|2
4. (7.25)
Taking η = 0, m = 0, and θ(x) = 2a〈x〉−β, we obtain
v(x) = −a〈x〉−β−4(β(β − 1)|x|2 − 3β) + a2β2〈x〉−2β−4|x|2. (7.26)
In the case of 0 ≤ β ≤ 1 and a > 0, we see that v ≥ 0 and v = O(〈x〉−β−2).
Furthermore −∆ + v has no non-positive eigenvalues. In the case of β > 1 and
a < 0, we see that however v 6≥ 0. We can estimate the number of non-positive
eigenvalues of −∆ + v by the Lieb-Thirring inequality [Lie76]:
# eigenvalues of −∆ + v ≤ 0 ≤ a3
∫|v−(x)|3/2dx, (7.27)
where v− denotes the negative part of v and a3 is a constant independent of v. This
yields that −∆+v has no non-positive eigenvalues for sufficiently small a. Thus the
lemma holds. 2
154
7.2 The Nelson model with a variable mass
7.2.1 Dirichlet forms and symmetric semigroups
Before going to study the Nelson Hamiltonian with variable coefficients we review
fundamental properties of Dirichlet forms and symmetric semigroups, which will be
used in the next sections. The general reference in this section is [Dav89].
We assume that the dimension of the configuration space is d. Let (E , D) be a
symmetric quadratic form on L2(Rd) with a form domain D. (E , D) is Markovian
if and only if for arbitrary ε > 0, there exists ρε such that
(1) ρε(t) = t for t ∈ [0, 1], −ε ≤ ρε(t) ≤ 1+ ε for all t ∈ R, 0 ≤ ρε(t)−ρε(s) ≤ t−sfor s < t,
(2) ρε f ∈ D and E (ρε f, ρε f) ≤ E (f, f) holds for f ∈ D.
A Markovian closed symmetric form (E , D) is called the Dirichlet form. When
C∞0 (Rd) is a form core of the Dirichlet form (E , D), it is called a regular Dirichlet
form. When f, g ∈ D satisfies suppf ∩ suppg = ∅, E (f, g) = 0. Then (E , D) is
called a local Dirichlet form.
Let gµν ∈ L1loc(Rd) and (gµν(x))1≤µ,ν≤d = g(x) satisfy
λ1(x)1l ≤ g(x) ≤ λ2(x)1l (7.28)
with strictly positive continuous functions λj. Define
Eg(f, g) =d∑
µ,ν=1
∫Rdgµν(x)∂νf(x)∂νg(x)dx (7.29)
for f, g ∈ C∞0 (Rd).
Proposition 7.3 Eg is closable quadratic form on C∞0 (Rd).
Proof: See [Dav89, Theorem 1.2.6]. 2
We denote the closure by Eg.
Proposition 7.4 Let L be the self-adjoint operator associated with Eg. Then
(i) e−tL, t ≥ 0, is contraction from Lp(Rd) to itself for all 1 ≤ p ≤ ∞,
155
(ii) e−tL, t ≥ 0, is positivity preserving.
Proof: See [Dav89, Theorem 1.3.5]. 2
Proposition 7.5 Suppose that K > 0 be a self-adjoint operator such that e−tK is
positivity preserving and e−tK is bounded on L∞(Rd). Let E denote the quadratic
form associated with K. Then
(1) A bound of the form
‖e−tKf‖∞ ≤ C1t−α/4‖f‖2
with α > 2 for all t ≥ 0 and all f ∈ L2(Rd) is equivalent to
‖f‖22α/(α−2) ≤ C2E (f, f).
(2) Suppose a bound ‖e−tKf‖∞ ≤ Ct‖f‖2 follows for all t ≥ 0 and all f ∈ L2(Rd).
Then e−tKf has an integral kernel e−tK(x, y) for all t ≥ 0 which satisfies that
0 ≤ e−tK(x, y) ≤ C2t/2 almost everywhere.
Proof: See [Dav89, Theorem 2.4.2] for (1), and [Dav89, Lemma 2.1.2] for (2). 2
Remark 7.6 Let K > 0 be a self-adjoint operator in L2(Rd) such that e−tK is
positivity preserving and e−tK is bounded on L∞(Rd). Then e−tK is bounded on
Lp(Rd) for 1 ≤ p ≤ ∞.
Suppose that L is the self-adjoint operator associated with the quadratic form Egdefined by (7.29) but λ1(x) and λ2(x) in (7.28) are replaced by positive constants
λ1 and λ2, respectively. Then L is called a strictly elliptic operator.
Proposition 7.7 Let L be a strictly elliptic operator. Then e−tL has an integral
kernel e−tL(x, y) and has Gaussian bounds:
C1eC2t∆(x, y) ≤ e−tL(x, y) ≤ C3e
C4t∆(x, y).
Proof: The upper and lower Gaussian bounds are proven in [Dav89, Corollary 3.2.8]
and [Dav89, Theorem 3.3.4], respectively. 2
156
7.2.2 Schrodinger operators with divergence form
We define the Schrodinger operator K on L2(R3) with variable coefficients. Let K0
be defined formally by
K0 =1
2
3∑µ,ν=1
DµAµν(x)Dν , (7.30)
where Dµ = −i∇µ with the domain D(Dµ) = H1(R3) describes the momentum
operator and A = A(x) = (Aµν(x))1≤µ,ν≤3 is a 3 × 3 symmetric matrix for each
x ∈ R3. We give the rigorous definition of K0 through a quadratic form. We
introduce an assumption on A(x).
Assumption 7.8 (Uniform elliptic condition) Suppose that each Aµν, 1 ≤µ, ν ≤ 3, is a measurable function, and A is uniformly elliptic, i.e., there exist
constants C0 > 0 and C1 > 0 such that
C01l ≤ A(x) ≤ C11l. (7.31)
Let E1l(f, g) and EA(f, g) be the quadratic forms defined by
EA(f, g) =1
2
3∑µ,ν=1
∫Aµν(x)∂µf(x) · ∂νg(x)dx (7.32)
and
E1l(f, g) =1
2
3∑µ=1
∫∂µf(x) · ∂µg(x)dx (7.33)
with the form domain H1(R3). Under Assumption 7.8, we have
C0E1l(f, f) ≤ EA(f, f) ≤ C1E1l(f, f), f ∈ H1(R3). (7.34)
From this inequality we can see that (EA, H1(R3)) is a closed semibounded form.
We define K0 by the unique self-adjoint operator associated with EA: there exists a
nonnegative self-adjoint operator K0 such that
EA(f, g) = (K1/20 f,K
1/20 g) (7.35)
with H1(R3) = D(K1/20 ). In general it is not easy to specify the operator domain of
K0. We can however specify it under some regularity conditions on Aµν(x). Let
W n,∞ =f ∈ L∞(R3)|∂zf ∈ L∞(R3) for |z| ≤ n
,
157
where ∂ denotes the distributional differential operator on L1loc(R3). It is fun-
damental that for f ∈ W 1,∞(R3) and u ∈ H1(R3), we have fu ∈ H1(R3) and
∂µ(fu) = (∂µf)u+ f∂µu for µ = 1, 2, 3.
Lemma 7.9 Suppose that each Aµν, 1 ≤ µ, ν ≤ 3, satisfies Aµν ∈ W 1,∞(R3), and
Assumptions 7.8. Then D(K0) = H2(R3) and
K0f =3∑
µ,ν=1
Dµ(Aµν(x)Dνf).
Proof: Since H2(R3) ⊂ D(K0) is trivial, it is enough to see H2(R3) ⊃ D(K0). Let
f ∈ K0 and T µt = eitDµ . Note that T µt f(x) = f(x + teµ), where eµ is the unit
vector in R3 to the µth direction, and DνTµt f = T µt Dνf follows for f ∈ H1(R3) with
µ, ν = 1, ..., d. It is a fundamental fact that f ∈ H1(R3) if and only if
sup0∈(0,1]
∥∥∥∥1
t(T µt − 1)f
∥∥∥∥L2
<∞, µ = 1, ..., d. (7.36)
Furthermore
sup0∈(0,1]
∥∥∥∥1
t(T µt − 1)f
∥∥∥∥ ≤ ‖f‖H1(R3), µ = 1, ..., d (7.37)
holds for f ∈ H1(R3). Then if f ∈ H1(R3) satisfies that
sup0∈(0,1]
∥∥∥∥1
t(T µt − 1)f
∥∥∥∥H1(R3)
<∞, µ = 1, ..., d, (7.38)
then f ∈ H2(R3). Let ‖f‖2EA
= ‖f‖2 + EA(f, f). We fix α = 1, ..., d, and set
∆tf(x) =1
t(Tαt − 1)f(x) =
1
t(f(x + teα)− f(x)).
Let f ∈ D(K0)(⊂ H1(R3)) and set ft = ∆tf . We will show that
supt∈(0,1]
‖ft‖H1(R3) <∞. (7.39)
We have ‖ft‖2EA
= (∆tf, ft)EA = Pt +Qt, where
Pt = −(f,∆−tft)L2 − (K0f,∆−tft)L2 ,
Qt = (f,∆−tft)EA + (∆tf, ft)EA .
158
We have
|Pt| ≤ ‖ft‖H1(R3)(‖f‖+ ‖K0f‖) ≤ ‖ft‖EA(‖f‖+ ‖K0f‖),
while
Qt = (f,∆−tft) + (∆tf, ft) +3∑
µ,ν=1
((AνµDµf,Dν∆−tft) + (AνµDµ∆tf,Dνft)) .
We have
(AνµDµf,Dν∆−tft) + (AνµDµ∆tf,Dνft) = (AνµDµ∆tf −∆t(AνµDνf), Dνft)
= (−∆tAνµ · Tt(Dµf), Dνft).
Then
|(AνµDµf,Dν∆−tft) + (AνµDµ∆tf,Dνft)| ≤ ‖∆tAνµ‖∞‖f‖H1(R3)‖ft‖H1(R3)
and
|Qt| ≤ C‖f‖H1(R3)‖ft‖EAfollows with some constant C independent of t. Then we see that
‖ft‖2EA≤ ‖ft‖H1(R3)(‖f‖+ ‖K0f‖) + C‖f‖H1(R3)‖ft‖EA
and
supt∈(0,1]
‖ft‖H1(R3) ≤ supt∈(0,1]
‖ft‖EA ≤ ‖f‖+ ‖K0f‖+ C‖f‖H1(R3) <∞.
Then (7.39) follows and the lemma is proven. 2
We furthermore introduce the assumption on external potentials W .
Assumption 7.10 (Confining potential) W ∈ L1loc(R3) and there exist δ > 0
and C > 0 such that
W (x) ≥ C〈x〉2δ. (7.40)
The Schrodinger operator K on L2(R3) with kinetic term K0 and an external po-
tential W satisfying Assumption 7.10 is defined by the quadratic form sum. Let
E (f, g) = EA(f, g) + (W 1/2f,W 1/2g) (7.41)
159
with the form domain C∞0 (R3). The quadratic form E is semibounded and then
closable. We denote the closure of E by the same symbol. We define K as the
unique self-adjoint operator associated with the quadratic form E :
E (f, g) = (K1/2f,K1/2g) (7.42)
for f, g in the quadratic form domain of E . We describe it as
K = K0 + W. (7.43)
Lemma 7.11 (Compact resolvent) Suppose Assumptions 7.8 and 7.10. Then K
has a compact resolvent and in particular it has a ground state.
Proof: In general a nonnegative self-adjoint operator T has a compact resolvent if
and only if
DT (b) = f ∈ D(T 1/2) | ‖f‖ < 1, ‖T 1/2f‖ ≤ b
is a compact set for all b > 0. See e.g., [RS78, Theorem XIII.64]. Let L = −12∆ +W .
Then L is essentially self-adjoint on C∞0 (R3) by Kato’s inequality, and since DL(b)
is compact for all b, L has a compact resolvent. By Assumption 7.8 we see that
‖L1/2f‖ ≤ C−10 ‖K1/2f‖ (7.44)
for f ∈ C∞0 (R3), where constant C0 is given by (7.31). By a limiting argument
(7.44) holds true for f ∈ D(K1/2), and DK(b) ⊂ DL(b/C0) follows. Then DK(b) is
compact for all b > 0, thus K has a compact resolvent. 2
In addition to Assumptions 7.8 and 7.10, suppose Assumption 7.18 (Lipshitz
condition) mentioned later. It will be proven in Corollary 7.25 that the normalized
ground state ϕp of K is strictly positive and unique. Define the probability measure
on R3 by
dµp = ϕ2p(x)dx (7.45)
and we set
Hp = L2(R3; dµp). (7.46)
We transform K by the ground state transformation for later use. Let
Up : Hp → L2(R3), f 7→ ϕpf.
160
Let Lp be the unitarily transformed operator of K − infσ(K) defined by
Lp = U−1p (K − infσ(K))Up (7.47)
with the domain D(Lp) = U−1p D(K). We note that
(f, Lpg)Hp = (ϕpf,Kϕpg)L2 − infσ(K)(ϕpf, ϕpg)L2 .
7.2.3 Scalar quantum fields
In the previous section we discuss the particle part. In the present section we
introduce a scalar quantum field. Let us begin with defining a scalar field in the
Schodinger representation. We use the notation EP for the expectation with respect
to a probability measure P , i.e.,∫· · · dP = EP [· · · ].
Let Q = SR(R3) be the set of real-valued rapidly decreasing and infinite-times
differentiable functions on R3. There exist a σ-field Σ, a probability measure µ on
(Q,Σ) and a Gaussian random variable φ(f) indexed by f ∈ L2R(R3) such that
Eµ[φ(f)] = 0 (7.48)
and the covariance given by
Eµ[φ(f)φ(g)] =1
2(f, g)L2 , (7.49)
and henceforth
Eµ[ezφ(f)
]= e(z2/4)‖f‖2 , z ∈ C. (7.50)
For general f ∈ L2(R3), φ(f) is defined by φ(f) = φ(<f) + iφ(=f). Thus φ(f) is
linear in f over C. The boson Fock space is defined by L2(Q, dµ) = L2(Q). The
identity function 1l ∈ L2(Q) is called the Fock vacuum. It is know that the linear
hull of
1l ∪ : φ(f1) · · ·φ(fn) : |fj ∈ L2(R3), j = 1, , ., n, n ≥ 1 (7.51)
161
is dense in L2(Q), where : φ(f1) · · ·φ(fn) : denotes the Wick product inductively
defined by
: φ(f) := φ(f),
: φ(f)n∏j=1
φ(fj) := φ(f) :n∏j=1
φ(fj) : −1
2
n∑k=1
(f, fk) :n∏j 6=k
φ(fj) : .
For a contraction operator T on L2(R3), define the second quantization Γ(T ) :
L2(Q)→ L2(Q) by Γ(T )1l = 1l and
Γ(T ) : φ(f1) · · ·φ(fn) :=: φ(Tf1) · · ·φ(Tfn) : . (7.52)
Then Γ(T ) is also contraction on (7.51) and can be uniquely extended to the con-
traction operator on the hole space L2(Q), which is denoted by the same symbol
Γ(T ). We can check that Γ(T )Γ(S) = Γ(TS). Then Γ(e−ith)t∈R for a self-adjoint
operator h defines the strongly continuous one-parameter unitary group on L2(Q).
The unique self-adjoint generator of Γ(e−ith)t∈R is denoted by dΓ(h), i.e.,
Γ(e−ith) = e−itdΓ(h), t ∈ R. (7.53)
7.2.4 The Nelson model with a variable mass
For the standard Nelson model the dispersion relation is given by (−∆ +m2)1/2
with
a constant m ≥ 0. In this note m is replaced by a positive function m(x) and −∆
by the divergence form∑3
µ,ν=1 c(x)−1Dµaµν(x)Dνc(x)−1. Let
h = h(D, x) =3∑
µ,ν=1
c(x)−1Dµaµν(x)Dνc(x)−1 +m2(x). (7.54)
In the same way as K0 we define h by the quadratic form, then the following as-
sumption is introduced.
Assumption 7.12 (Condition on ω) Let a = a(x) = (aµν(x))1≤µ,ν≤3.
(1) (Uniform elliptic condition) aµν ∈ W 1,∞ and there exist constants C0 > 0
and C1 > 0 such that
C01l ≤ a(x) ≤ C11l. (7.55)
162
(2) (Uniform bound) There exist 0 < C0 and 0 < C1 such that C0 ≤ c(x) ≤ C1
and c ∈ W 2,∞.
(3) (Decay of the variable mass) There exists β > 2 such that
m(x) ≤ 〈x〉−β/2. (7.56)
Under Assumption 7.12 let us define the semibounded quadratic form
(f, g) 7→ Ea(f, g)
=3∑
µ,ν=1
∫aµν(x)∂µ
(1
c(x)f(x)
)· ∂ν
(1
c(x)g(x)
)dx+
∫m2(x)f(x)g(x)dx
for f, g ∈ H1(R3), which is closable. Notice that c−1f ∈ H1(R3) if f ∈ H1(R3),
since c−1 ∈ W 2,∞, and ∂µ(c−1f) = ∂c−1 · f + c−1 · ∂µf .
Definition 7.13 (Dispersion relation with a variable mass) Operator h is
defined by the nonnegative self-adjoint operator associated with the closure of Ea,
and the self-adjoint operator ω on L2(R3) is defined by
ω = h1/2. (7.57)
Lemma 7.14 Suppose Assumption 7.12. Then h is self-adjoint on H2(R3), and
infσ(h) = 0 but 0 is not an eigenvalue of h. In particular Ker ω = 0.
Proof: Directly we can see that c−1f ∈ H2(R3) if f ∈ H2(R3) and
hf = h0f + vf, (7.58)
where
h0f =3∑
µ,ν=1
Dµ(c−1aµνc−1Dνf), (7.59)
and, by assumptions c ∈ W 2,∞ and aµν ∈ W 1,∞, v is the bounded multiplication
operator given by
v = m2 +3∑
µ,ν=1
(c−1(∂νaµν)(∂µc
−1) + c−1aµν(∂µ∂νc−1)).
163
Since D(h0) = H2(R3) by c−1aµνc−1 ∈ W 1,∞, h is self-adjoint on H2(R3) by the
Kato-Rellich theorem. By (7.58) we have
Ea(f, f) ≤ D1E1l(c−1f, c−1f) + (mf,mf), (7.60)
D2E1l(c−1f, c−1f) + (mf,mf) ≤ Ea(f, f) (7.61)
with some constants D1 and D2. Notice that −Dj∆ + m2 has no zero eigenvector
and σ(−Dj∆ + m2) = [0,∞), since m2 is a compact perturbation of −C∆. By
(7.60), h has also no zero eigenvector and infσ(h) = 0. 2
Definition 7.15 (The Nelson Hamiltonian with a variable mass)
The Nelson Hamiltonian with a variable mass m(x) and a cutoff function ρ is
defined by
H = Lp ⊗ 1l + 1l⊗Hf + φρ (7.62)
on the tensor product Hilbert space H = Hp ⊗ L2(Q), where we set the coupling
constant α as α = 1, Lp is defined by (7.47), the free field Hamiltonian Hf by
Hf = dΓ(ω) and the field operator φρ is given by
φρ =
∫ ⊕R3
φρ(x)dµp (7.63)
with φρ(x) = φ(ω−1/2ρ(· − x)). Here we used the identification H ∼=∫ ⊕R3 L
2(Q)dµp.
Thus the Nelson Hamiltonian is a linear operator defined on the L2-space over the
probability space (R3 ×Q, dµp ⊗ dµ).
Assumption 7.16 (Condition on ρ) The ultraviolet cutoff function ρ satisfies
that
(1) ρ ≥ 0, (2) ρ/√|k| ∈ L2(R3), (3) ρ/|k| ∈ L2(R3). (7.64)
We will use (1) of Assumption 7.16 in the proof of Lemma 7.42.
Proposition 7.17 Suppose Assumptions 7.8, 7.10, 7.12 and 7.16. Then the Nelson
Hamiltonian H is self-adjoint on D(Lp)∩D(Hf), and bounded from below. Further-
more it is essentially self-adjoint on any core of Lp +Hf .
164
Proof: By Assumption 7.12 it follows that
supx‖ω−n/2ρ(· − x)‖ ≤ C‖ρ/|k|n/2‖ (7.65)
for n = 1, 2 with some C. (7.65) is shown in Corollary 7.40 later. Then φρ(x) is
infinitesimally small with respect to Hf for each x ∈ R3. Then φρ is infinitesimally
small with respect to Lp + Hf , and the proposition follows by the Kato-Rellich
theorem. 2
7.3 Feynman-Kac formula and diffusions
In this section we construct a functional integral representation of the one-parameter
heat semigroup e−tH .
7.3.1 Super-exponential decay
The following assumption ensures the existence and uniqueness of a stochastic dif-
ferential equation. Let Cnb (R3) = f ∈ Cn(R3)|fm ∈ L∞(R3), |m| ≤ n.
Assumption 7.18 (Lipshitz condition) Suppose that Aµν ∈ C1b(R3), µ, ν =
1, 2, 3, and bν(x) =1
2
3∑µ=1
∂µAµν(x) and the 3 × 3 matrix (σµν(x))1≤µ,ν≤3 = σ(x) =√A(x) satisfy the Lipshitz condition:
|b(x)− b(y)|+ |σ(x)− σ(y)| ≤ D|x− y| (7.66)
for arbitrary x, y ∈ R3 with some constant D independent of x and y, where |σ(x)| =√∑3µ,ν=1 |σµν(x)|2.
Lemma 7.19 Suppose Assumptions 7.8 and 7.18. Then D(K0) = H2(R3) and
K0f =∑3
µ,ν=1Dµ(AµνDνf) for f ∈ H2(R3).
Proof: We see that Aµν ∈ W 1,∞(R3). Then the lemma immediately follows from
Lemma 7.9. 2
Let us consider the stochastic differential equation:dXν
t = σν(Xt) · dBt + bν(Xt)dt,Xν
0 = xν ,(7.67)
165
on the probability space (X+,B(X+),W), where we recall X+ = C([0,∞);R3),
B(X+) is the σ-field generated by cylinder sets andW the Wiener measure starting
at 0. We denote EW by E unless confusions may arise. (Bt)t≥0 denotes the 3-
dimensional Brownian motion on (X+,B(X+),W). The drift term bν and the
diffusion term σν = (σν1, σν2, σν3) are defined in Assumption 7.18. Note that bν and
σµν are bounded; ‖bν‖∞ <∞ and ‖σµν‖∞ <∞. Let (Ft)t≥0 be the natural filtration
of the Brownian motion: Ft = σ(Bs, 0 ≤ s ≤ t).
Proposition 7.20 Suppose Assumption 7.18. Then (7.67) has the unique solution
Xx = (Xxt )t≥0 which is a diffusion process with respect to the filtration (Ft)t≥0.
Namely Xx has continuous sample paths and Markov property:
E[f(Xx
s+t)|Fs]
= E[f(X
Xxs
t )], (7.68)
where E[f(X
Xxs
t )]
is E [f(Xyt )] evaluated at y = Xx
s .
From (7.68) we can show that
Ttf(x) = E [f(Xxt )] (7.69)
satisfies the semigroup property TsTtf = Ts+tf on L∞(R3). In the next proposition
we show indeed that Ttf ∈ L2(R3) for f ∈ L2(R3). Namely Tt defines a semigroup
not only on L∞(R3) but also on L2(R3).
In order to show that Tt : L∞ → L∞ can be extended to a semigroup on L2(R3),
we introduce a Dirichlet form. Suppose Assumption 7.8. We see that (EA, H1(R3))
is a local and regular Dirichlet form . It is a fundamental fact that there exist a
probability measure νx on (X+,B(X+)) and a coordinate process Z = (Zt)t≥0 such
that (1) νx(Z0 = x) = 1, (2) Z is a symmetric diffusion process with respect to the
natural filtration Mt = σ(Zs, 0 ≤ s ≤ t), (3)
Stf(x) = Eνx [f(Zt)] (7.70)
defines the semigroup, and (4) for each t ≥ 0,(e−tK0f
)(x) = (Stf) (x), a.e. x ∈ R3. (7.71)
See e.g.,[Fuk80, Lemma 4.3.1].
166
Proposition 7.21 (L2 extension) Let f ∈ L2(R3) ∩ L∞(R3). Suppose Assump-
tions 7.8 and 7.18. Then
Ttf = e−tK0f, t ≥ 0, a.e. (7.72)
In particular TtdL2∩L∞ = e−tK0, where · · · denotes the closure in L2(R3).
Proof: Let f ∈ C∞0 (R3). We set
Mt = f(Zt)−∫ t
0
(K0f)(Zs)ds,
Nt = f(Xxt )−
∫ t
0
(K0f)(Xxs )ds.
By the Ito formula we have
f(Xxt )− f(x) =
∫ t
0
(K0f)(Xxs )ds+
3∑µ=1
∫ t
0
(∂µf)(Xxs )σµ(Xx
s ) · dBs.
Hence
Nt = f(x) +3∑
µ=1
∫ t
0
(∂µf)(Xxs )σµ(Xx
s ) · dBs
and then (Nt)t≥0 is martingale on (X+,B(X+),W) with respect to (Ft)t≥0, while
we can see that
Eνx [Mt+s|Zs] = E[f(Zt+s)|Zs]−∫ s
0
K0f(Zr)dr − E[∫ t+s
s
K0f(Zr)dr |Zs
].
Here Zt = σ(Zs, 0 ≤ s ≤ t). Let p(t, y, A) be the probability transition kernel of Ztunder νx. Then by the Markov property of Z we have
E[f(Zt+s)|Zs] =
∫f(y)p(t, Zs, dy) = (e−tK0f)(Zs)
and
E[∫ t+s
s
(K0f)(Zr)dr |Zs
]=
∫ t+s
s
dr
∫(K0f)(y)p(r − s, Zs, dy)
=
∫ t
0
dr
∫(K0f)(y)p(r, Zs, dy) =
∫ t
0
(e−rK0K0f)(Zs)dr = (e−tK0f)(Zs)− f(Zs).
167
Then we see that
E[Mt+s|Zs] = f(Zs)−∫ s
0
K0f(Zr)dr = Ms
and we conclude that (Mt)t≥0 is also martingale with respect to (Zt)t≥0. By the
uniqueness of martingale problem (e.g., [RW00, Theorem (24.1)], [SV06, Chapter
8],[KS91, Section 5.4.E]), it follows that νx =W (Xx)−1. In particular
Eνx [f(Zt)] = EW [f(Xxt )],
which is equivalent to Ttf = Stf . Then the proposition follows from (7.71). 2
In order to see some properties of e−tK0 , we give a Gaussian bound of the integral
kernel of e−tK0 . When ‖e−tLf‖∞ ≤ Ct‖f‖2 is satisfied for all t > 0 and all f ∈L2(R3), e−tL is called ultracontractivity. .
Proposition 7.22 (Kernels) Suppose Assumption 7.8. Then e−tK0 is ultracon-
tractive, has an integral kernel, and the kernel satisfies that
C1etC2∆(x, y) ≤ e−tK0(x, y) ≤ C3e
tC4∆(x, y) (7.73)
with some constants Cj, j = 1, 2, 3, 4, where
eT∆(x, y) = (2πT )−3/2 exp(−|x− y|2/(2T ))
is the 3-dimensional heat kernel.
Proof: See Propositions 7.4, 7.5 and 7.7. 2
We prove the Feynman-Kac formula of e−t(K0+W ) for general W . Let h0 =
(−1/2)∆. Suppose that W is form bounded with respect to h0 with a relative bound
b, i.e.,
limE→∞
‖|W |1/2(h0 + E)−1/2f‖/‖f‖ = b.
By Proposition 7.22 we notice that
|(f, e−tK0g)| ≤ (|f |, C ′e−tCh0|g|), (7.74)
168
where C ′ and C are nonnegative constants. Let T be nonnegative self-adjoint oper-
ator. Then (T + E)−1/2 = π−1/2∫∞
0t−1/2e−(T+E)dt for E > 0. From this formula and
(7.74) it follows that
|(K0 + E)−1/2f |(x) ≤ C ′(Ch0 + E)−1/2|f |(x). (7.75)
Hence
‖|W |1/2(K0 + E)−1/2f‖/‖f‖ ≤ C ′‖|W |1/2(Ch0 + E)−1/2|f |‖/‖f‖ (7.76)
and we have
limE→∞
‖|W |1/2(K0 + E)−1/2f‖/‖f‖ = C ′C−1/2b.
Then W is also relatively form bounded with respect to K0 with a relative bound
< C ′C−1/2b. We introduce an assumption on W .
Assumption 7.23 Let W = W+ −W−, where W± = max±W, 0. Suppose W+ ∈L1
loc(Rd) and W− is relatively form bounded with respect to h0 with a relative bound
b such that
C ′C−1/2b < 1, (7.77)
where constants C,C ′ are in (7.74).
Suppose Assumptions 7.8 and 7.23. Then by the KLMN theorem
K = K0 + W+ − W− (7.78)
can be defined as a self-adjoint operator. Here ± denotes the quadratic form sum.
Proposition 7.24 (Feynman-Kac formula) Suppose Assumptions 7.8, 7.18 and
7.23. Let K be given by (7.78). Then
(g, e−tKf
)=
∫dµpE
[g(x)f(Xx
t )e−∫ t0 W (Xx
s )ds]. (7.79)
In particular (e−tKf
)(x) = E
[f(Xx
t )e−∫ t0 W (Xx
s )ds]. (7.80)
169
Proof: Suppose first that W is bounded and continuous. By the Trotter-Kato prod-
uct formula we have
(f, e−tKg) = limn→∞
(f, (e−(t/n)K0e−(t/n)W )ng). (7.81)
By Proposition 7.21 we have for t0 ≤ t1 ≤ · · · ≤ tn,(e−(t1−t0)K0f1 · · · e−(tn−tn−1)K0fn
)(x)
= E[f1(Xx
t1−t0)(e−(t2−t1)K0f1 · · · e−(tn−tn−1)K0fn
)(Xx
t1−t0)].
By the Markov property (7.68) we also have
= E[f1(Xx
t1−t0)E[f2(X
Xxt1−t0
t2−t1 )(e−(t3−t2)K0f1 · · · e−(tn−tn−1)K0fn
)(X
Xxt1−t0
t2−t1 )]]
= E[f1(Xx
t1−t0)E[f2(Xx
t2−t0)(e−(t3−t2)K0f1 · · · e−(tn−tn−1)K0fn
)(Xx
t2−t0) |Ft1−t0]]
= E[f1(Xx
t1−t0)f2(Xxt2−t0)
(e−(t3−t2)K0f1 · · · e−(tn−tn−1)K0fn
)(Xx
t2−t0)].
Inductively we obtain that
(e−(t1−t0)K0f1 · · · e−(tn−tn−1)K0fn
)(x) = E
[n∏j=1
fj(Xxtj−tj−1
)
]. (7.82)
Combining the Trotter product formula (7.81) and (7.82) with tj = tj/n, we have
(f, e−tKg) = limn→∞
∫dxf(x)E
[e−(t/n)
∑nj=1W (Xx
tj/n)g(Xx
t )]. (7.83)
Since s 7→ Xxs (ω) has continuous paths, W (Xx
s (ω)) is continuous in s ∈ [0, t] for
each ω. Therefore∑n
j=1tnW (Xx
tj/n)→∫ t
0W (Xx
s )ds as n→∞ for each ω and exists
as a Riemann integral.
In order to extend W to more general class, we use a standard limiting argument.
To do that, suppose that W ∈ L∞ and Wn(x) = φ(x/n)(W ∗ jn)(x), where jn =
n3φ(xn) with φ ∈ C∞0 (R3) such that 0 ≤ φ ≤ 1,∫φ(x)dx = 1 and φ(0) = 1.
Then Wn is bounded and continuous, moreover Wn(y) → W (y) as n → ∞ for
y 6∈ N with some null set N . Notice that E[1lXxt ∈N ] =
∫1ly∈N e−tK0(x, y)dy = 0
and thus∫ t
0dsE[1lXx
s∈N ] = E[∫ t
0ds1lXx
s∈N
]= 0 by Fubini’s lemma. Thus for each
170
x ∈ R3 the measure of t ∈ [0,∞) |Xxt (ω) ∈ N is zero for almost every ω. Hence∫ t
0Wn(Xx
s )ds→∫ t
0W (Xx
s )ds as n→∞ almost surely, and∫dµpE
[f(x)g(Xx
t )e−∫ t0 Wn(Xx
s )ds]→∫dµpE
[f(x)g(Xx
t )e−∫ t0 W (Xx
s )ds]
as n → ∞. On the other hand, e−t(K0+Wn) → e−t(K0+W ) strongly as n → ∞, since
K0 +Wn converges to K0 +W on the common domain H2(R3). Next define
W+,n(x) =
W+(x), W+(x) < n,n, W+(x) ≥ n,
W−,m(x) =
W−(x), W−(x) < m,m, W−(x) ≥ m.
Note that Q(K0) = H1(R3), where Q(T ) denotes the form domain of T , i.e., Q(T ) =
D(|T |1/2). Define the closed quadratic forms
qn,m(f, f) = (K1/20 f,K
1/20 f) + (W
1/2+,nf,W
1/2+,nf)− (W
1/2−,mf,W
1/2−,mf),
qn,∞(f, f) = (K1/20 f,K
1/20 f) + (W
1/2+,nf,W
1/2+,nf)− (W
1/2− f,W
1/2− f),
q∞,∞(f, f) = (K1/20 f,K
1/20 f) + (W
1/2+ f,W
1/2+ f)− (W
1/2− f,W
1/2− f),
where the form domains are given by
Q(qn,m) = H1(R3), Q(qn,∞) = H1(R3), Q(q∞,∞) = H1(R3) ∩Q(W+).
Note that
qn,m ≥ qn,m+1 ≥ qn,m+2 ≥ ... ≥ qn,∞
and qn,m → qn,∞ in the sense of quadratic forms on ∪mQ(qn,m) = H1(R3). Since
qn,∞ is closed on H1(R3), by the monotone convergence theorem for a non-increasing
sequence of forms (see [Kat76, VIII. Theorem 3.11])the associated positive self-
adjoint operators satisfy K0 + W+,n − W−,m → K0 + W+,n − W− in strong resol-
vent sense, which implies that
e−t(K0 + W+,n − W−,m) → e−t(K0 + W+,n − W−) (7.84)
strongly as m→∞ for all t ≥ 0. Similarly, we have
qn,∞ ≤ qn+1,∞ ≤ qn+2,∞ ≤ ... ≤ q∞,∞
171
and qn,∞ → q∞,∞ in quadratic form sense on
f ∈ ∩nQ(qn,∞) | supn
qn,∞(f, f) <∞ = H1(R3) ∩Q(W+).
Hence by the monotone convergence theorem for a non-decreasing sequence of forms
(see [Kat76, VIII. Theorem 3.13 and p.575]) we obtain
e−t(K0 + W+,n − W−) → e−t(K0 + W+ − W−), (7.85)
strongly as n→∞. On the other hand, we can see that∫dxE
[e−
∫ t0 (W+,n−W−,m)(Xx
s )ds]−→
∫dxE
[e−
∫ t0 (W+,n−W−)(Xx
s )ds]
as m→∞. Moreover,∫dxE
[e−
∫ t0 (W+,n−W−)(Xx
s )ds]−→
∫dxE
[e−
∫ t0 (W+−W−)(Xx
s )ds]
as n → ∞, by (7.85) and the dominated convergence theorem. Thus the proof is
complete. 2
Corollary 7.25 (Positivity improving) Suppose Assumptions 7.8, 7.10 and 7.18.
Then e−tK is positivity improving. In particular the ground state of K is strictly
positive and unique.
Proof: Let f ≥ 0 and g ≥ 0 but f 6≡ 0 and g 6≡ 0. It is enough to show that
(f, e−tKg) > 0. Let suppf = Df and suppg = Dg. We first show that for each
x ∈ R3,
W(∫ t
0
W (Xxs )ds =∞
)= 0. (7.86)
Let us recall that (Bt)t≥0 is the Brownian motion on (X+,B(X+),W). Let N ∈ N.
Since W ∈ L1loc(R3), 1lNW ∈ L1(R3) and then by Proposition 7.22∫dxE
[∫ t
0
1lNW (Xxs )ds
]=
∫dx
∫ t
0
E [1lNW (Xxs )] ds
≤∫dxC3
∫ t
0
E [1lNW (B2C4t + x)] ds ≤ ‖1lNW‖L1C3t <∞.
172
Thus W(∫ t
01lNW (Xx
s )ds <∞) = 1 and there exists NN such that W(NN) = 0 and∫ t0
1lNW (Xxs (ω))ds <∞ for arbitrary ω ∈X+ \NN . Let N = ∪NNN . Thus∫ t
0
1lNW (Xxs (ω))ds <∞
for arbitrary N ∈ N and ω ∈ X+ \N . Since Xxs is continuous in s, for each ω ∈
X+ \N , there exists N = N(ω) such that sup0≤s≤tXxs (ω) < N . Then W (Xx
s (ω)) =
1lNW (Xxs (ω)) for 0 ≤ s ≤ t, and∫ t
0
W (Xxs (ω))ds =
∫ t
0
1lNW (Xxs (ω))ds <∞.
This implies (7.86) and e−∫ t0 W (Xx
s (ω))ds > 0 for a.e. ω ∈ X+. By the Feynman-Kac
formula, it is sufficient to see that∫dxE [f(Xx
0 )g(Xxt )] > 0. Let
Dxg = ω ∈X+|Xx
t (ω) ∈ Dg.
Thus ∫Df
dxE[1lDx
g
]= (1lDf , e
−tK01lDg) ≥ C1(1lDf , eC2t∆1lDg) > 0.
Then the measure of ∪x∈DfDxg(⊂ R3×X+) is strictly positive with respect to dx⊗dW
and f(Xx0 )g(Xx
t ) > 0 on ∪x∈DfDxg . Then∫
dxE [f(Xx0 )g(Xx
t )] ≥∫Df
dx
∫Dxg
f(Xx0 )g(Xx
t )dW > 0
and the corollary follows. 2
Corollary 7.26 (Ultracontractivity) Suppose Assumptions 7.8, 7.10 and 7.18.
Then e−tK is ultracontractive.
Proof: Note that e−∫ t0 W (Xx
s )ds ≤ 1. By the Feynman-Kac formula, we have∣∣(e−tKf) (x)∣∣ ≤ (E [|f(Xx
t )|2])1/2
.
By Proposition 7.22 we have
E[|f(Xx
t )|2]
=(e−tK0|f |2
)(x) ≤ C3
(eC4t∆|f |2
)(x) ≤ Ct−3/2‖f‖2
L2 .
Then ‖e−tKf‖∞ ≤ Ct−3/4‖f‖L2 and the corollary follows. 2
We can also prove the theorem below.
173
Theorem 7.27 (Super-exponential decay) Suppose Assumptions 7.8, 7.10 and
7.18. Then there exists a constant γ > 0 such that
eγ|x|δ+1
ϕp ∈ H1(R3). (7.87)
Proof: Let F ∈ C∞(R3) be real, bounded with all derivatives. Then for u ∈ D(K)
we have the Agmon identity:∫1
2∇(eF u) · A∇(eFu)dx +
∫e2F (W − 1
2∇F · A∇F )|u|2dx
=
∫e2F uKudx + 2iIm
∫e2F∇u · A∇Fdx.
Applying this identity to the ground state ϕp, we obtain that eγ〈x〉δ+1ϕp ∈ L2(R3)
and ∇(eγ〈x〉δ+1ϕp) ∈ L2(R3). 2
7.3.2 Diffusion processes
We can also construct a Markov process X = (Xt)t∈R on the hole real line R asso-
ciated with the semigroup e−tLp by a stochastic differential equation. Let
X = C(R;R3).
Proposition 7.28 (Diffusion process associated with e−tLp) Let Xt(ω) = ω(t)
be the coordinate process on (X ,B(X )). Suppose Assumptions 7.8, 7.10 and 7.18.
Then there exists a probability measure P x on (X ,B(X )) satisfying (1)-(4) below:
(1) (Initial distribution) P x(X0 = x) = 1.
(2) (Reflection symmetry) Two processes (Xt)t≥0 and (Xs)s≤0 are independent
and X−td= Xt.
12
12 Xd= Y means that X and Y has the same distribution.
174
(3) (Diffusion property) Let (F+t )t≥0 = σ(Xs, 0 ≤ s ≤ t) and (F−t )t≤0 =
σ(Xs,−t ≤ s ≤ 0) be filtrations. Then (Xt)t≥0 and (Xs)s≤0 are diffusion
processes with respect to (F+t )t≥0 and (F−t )t≤0, respectively, i.e.,
EP x [Xt+s|F+s ] = EP x [Xt+s|σ(Xs)] = EPXs [Xt],
EP x [X−t−s|F−−s] = EP x [X−t−s|σ(X−s)] = EPX−s [X−t]
for s, t ≥ 0, and Xt is continuous in t ∈ R, where EPXs means EP y evaluated
at y = Xs.
(4) (Shift invariance) It follows that∫dµpEP x [f0(Xt0) · · · fn(Xtn)] = (f0, e
−(t1−t0)Lpf1 · · · e−(tn−tn−1)Lpfn)Hp
(7.88)
for fj ∈ L∞(R3), j = 1, ..., n, and then the finite dimensional distribution of
X is shift-invariant, i.e.,∫dµpEP x
[n∏j=1
fj(Xtj)
]=
∫dµpEP x
[n∏j=1
fj(Xtj+s)
], s ∈ R,
for any bounded Borel measurable functions fj, j = 1, ..., n.
In order to prove Proposition7.28 we need several steps. An outline of constructing
a diffusion process (Xt)t∈R is as follows.
For 0 ≤ t0 ≤ t1 ≤ · · · ≤ tn let the set function νt0,...,tn :∏n
j=0 B(R3) → R be
given by
νt0,...,tn
(n∏i=0
Ai
)= (1lA0 , e
−(t1−t0)Lp1lA1 · · · e−(tn−tn−1)Lp1lAn)Hp (7.89)
and for 0 ≤ t, νt : B(R3)→ R by
νt (A) = (1l, e−tLp1lA)Hp = (1l, 1lA)Hp , (7.90)
where B(R3) denotes the Borel σ-field of R3. We show an outline of steps of the
proof.
175
(Step 1) By the Kolmogorov extension theorem we can construct a probability measure
ν∞ on (R3)[0,∞) from the family of probability measures given by (7.89) and
(7.90), and we define a stochastic process Y = (Yt)t≥0 on a probability space
((R3)[0,∞),B((R3)[0,∞)), ν∞) such that finite dimensional distributions of Y is
given by the right-hand side of (7.89) and (7.90). We also show the existence
of the continuous version (Yt)t≥0 on the same probability space.
(Step 2) Let Q = ν∞ Y −1 be the image measure of ν∞ on (X+,B(X+)), where
X+ = C([0,∞);R3). Let (Yt)t≥0 be the coordinate process on the probability
space (X+,B(X+), Q), i.e., Yt(ω) = ω(t) for ω ∈X+. Notice that Yd= Y .
(Step 3) Define a regular conditional probability measure by Qx(·) = Q(·|Y0 = x). Then
the stochastic process (Yt)t≥0 on a probability space (X+,B(X+), Qx) satisfies
(f0, e−(t1−t0)Lpf1 · · · e−(tn−tn−1)Lpfn)Hp =
∫dµpEQx
[n∏j=0
fj(Ytj)
](7.91)
for 0 ≤ t0 ≤ t1 ≤ · · · ≤ tn and we can show that Y is a diffusion process with
respect to the natural filtration σ(Ys, 0 ≤ s ≤ t).
(Step 4) We extend Y to a process of the hole real line. Define a stochastic process
Xt(ω) =
Yt(ω1), t ≥ 0,
Y−t(ω2), t < 0on the product probability space (X+, M , Qx) =
(X+ ×X+,B(X+)×B(X+), Qx ×Qx). This is a continuous process.
(Step 5) We will prove Proposition 7.28 in this step. Let P x be the image measure given
by P x = QxX−1 on (X ,B(X )), where X = C(R;R3). Then the coordinate
process (Xt)t∈R on the probability space X = (X ,B(X ), P x) satisfies that∫dµpEP x [f0(Xt0) · · · fn(Xtn)] = (f0, e
−(t1−t0)Lpf1 · · · e−(tn−tn−1)Lpfn)Hp
(7.92)
for −∞ < t0 ≤ t1 ≤ · · · ≤ tn. By this we can see that X = (Xt)t∈R satisfies
(1)-(4) of Proposition 7.28.
176
(Step 1) The family of set functions νξξ⊂R,#ξ<∞ given by (7.89) and (7.90)
satisfies the consistency condition:
νt0,...,tn+m
(n∏i=0
Ai ×n+m∏i=n+1
R3
)= νt0,...,tn
(n∏i=0
Ai
)and by the Kolmogorov extension theorem [KS91, Theorem 2.2] there exists a a
probability measure ν∞ on ((R3)[0,∞),B((R3)[0,∞))) such that
νt (A) = Eν∞ [1lA(Yt)] , (7.93)
νt0,...,tn
(n∏i=0
Ai
)= Eν∞
[n∏j=0
1lAj(Ytj)
], n ≥ 1, (7.94)
where Yt(ω) = ω(t), ω ∈ (R3)[0,∞), is the coordinate process. Then the process
Y = (Yt)t≥0 on the probability space ((R3)[0,∞),B((R3)[0,∞)), ν∞) satisfies that
(f0, e−(t1−t0)Lpf1 · · · e−(tn−tn−1)Lpfn)Hp = Eν∞
[n∏j=0
fj(Ytj)
], (7.95)
(1l, f)Hp = (1l, e−tLpf)Hp = Eν∞ [f(Yt)] = Eν∞ [f(Y0)] (7.96)
for fj ∈ L∞(R3), j = 0, 1, ..., n.
(Step 2) We now see that the process Y has a continuous version.
Lemma 7.29 The process Y on ((R3)[0,∞),B((R3)[0,∞)), ν∞) has a continuous ver-
sion.
Proof: We note that by (7.95), (7.96) and Proposition 7.24, Eν∞ [|Yt − Ys|2n] can be
expressed in terms of the diffusion process X = (Xxt )t≥0. Since
Eν∞ [|Yt − Ys|2n] =2n∑k=0
[2nk
](−1)kEν∞
[Y 2n−kt Y k
s
],
the left hand side above can be expressed in terms of e−tLp as
Eν∞ [|Yt − Ys|2n] =2n∑k=0
[2nk
](−1)k
(x2n−k, e−(t−s)Lpxk
)Hp
=2n∑k=0
[2nk
](−1)k
(x2n−kϕp, e
−(t−s)Kxkϕp
)L2 e
(t−s) infσ(Lp).
177
Furthermore by Feynman-Kac formula, i.e., Proposition 7.24, the right-hand side
above can be expressed in terms of Xx = (Xxt )t≥0 as
Eν∞ [|Yt − Ys|2n]
=
∫dµpE
[|Xx
t−s −Xx0 |2nϕp(Xx
0 )ϕp(Xxt−s)e
−∫ t−s0 W (Xx
r )dr]e(t−s) infσ(Lp).
Since W ≥ 0,
Eν∞ [|Yt − Ys|2n] ≤ ‖ϕp‖2∞e
(t−s) infσ(Lp)
∫dµpE
[|Xx
t−s −Xx0 |2n
].
We next estimate E [|Xxt −Xx
s |2n]. Since Xxt is the solution to the stochastic differ-
ential equation:
Xx,µt −Xx,µ
s =
∫ t
s
bµ(Xxr )dr +
∫ t
s
σµ(Xxr ) · dBr,
we have
E[|Xx,µ
t −Xx,µs |2n
]≤ 22n−1E
[|t− s|2n
22n‖bµ‖2n
∞ +3∑
ν=1
∣∣∣∣∫ t
s
σµν(Xxr )dBν
r
∣∣∣∣2n].
By the Burkholder-Davies-Gundy inequality [KS91, Theorem 3.28], we have
E
[∣∣∣∣∫ t
s
σµν(Xxr )dBν
r
∣∣∣∣2n]≤ (n(2n− 1))n|t− s|n−1E
[∫ t
s
|σµν(Xxr )|2ndr
]≤ (n(2n− 1))n|t− s|n‖σµν‖2n
∞ .
Then E [|Xxt −Xx
s |2n] ≤ C|t−s|n with some constant C independent of s and t, and
Eν∞[|Yt − Ys|2n
]≤ C|t− s|n (7.97)
follows. Thus Y = (Yt)t≥0 has a continuous version by Kolmogorov-Centov continu-
ity theorem [KS91, Theorem 2.8]. 2
Let Y = (Y t)t≥0 be the continuous version of Y on ((R3)[0,∞),B((R3)[0,∞)), ν∞).
The image measure of ν∞ on (X+,B(X+)) with respect to Y is denoted by Q, i.e.,
Q = ν∞ Y−1
, and Yt(ω) = ω(t) for ω ∈ X+ is the coordinate process. Then we
178
constructed a stochastic process Y = (Yt)t≥0 on (X+,B(X+), Q) such that Yd= Y .
Then (7.95) and (7.96) can be expressed in terms of Y as
(f0, e−(t1−t0)Lpf1 · · · e−(tn−tn−1)Lpfn)Hp = EQ
[n∏j=0
fj(Ytj)
],
(1l, f)Hp = (1l, e−tLpf)Hp = EQ[f(Yt)
]= EQ
[f(Y0)
]for 0 ≤ t and 0 ≤ t0 ≤ t1 ≤ · · · ≤ tn.
(Step 3) Define the regular conditional probability measure on X+ by
Qx(·) = Q(·|Y0 = x) (7.98)
for each x ∈ R3. It is well defined, since X+ is a Polish space (completely separable
metrizable space). See e.g., [KS91, Theorems 3.18. and 3.19]. Since the distribution
of Y0 equals to dµp, note that Q(A) =∫dµpEQx [1lA]. Then the stochastic process
Y = (Yt)t≥0 on (X+,B(X+), Qx) satisfies
(f0, e−(t1−t0)Lpf1 · · · e−(tn−tn−1)Lpfn)Hp =
∫dµpEQx
[n∏j=0
fj(Ytj)
], (7.99)
(1l, e−tLpf)Hp = (1l, f)Hp =
∫dxϕ2
p(x)EQx
[f(Y0)
]=
∫dxϕ2
p(x)f(x). (7.100)
Lemma 7.30 Y is a Markov process on (X+,B(X+), Qx) with respect to the nat-
ural filtration (Mt)t≥0, where Ms = σ(Yr, 0 ≤ r ≤ s).
Proof: Let
pt(x, A) =(e−tLp1lA
)(x), A ∈ B(R3), t ≥ 0. (7.101)
Notice that pt(x, A) = E [1lA(Xxt )]. Then the finite dimensional distribution of Y is
EQx
[n∏j=1
1lAj(Ytj)
]=
∫ n∏j=1
1lAj(xj)n∏j=1
ptj−tj−1(xj−1, dxj)
with t0 = 0 and x0 = x by (7.99). We show that pt(x, A) is a probability transition
kernel, i.e., (1) pt(x, ·) is a probability measure on B(R3), (2) pt(x, A) is Borel
measurable with respect to x, (3) the Chapman-Kolmogorov equality∫p(s, y, A)p(t, x, dy) = p(s+ t, x, A) (7.102)
179
is satisfied. Note that e−tLp is positivity improving. Then 0 ≤ e−tLpf ≤ 1l for all
function f such that 0 ≤ f ≤ 1l, and e−tLp1l = 1l follows. Then pt(x, ·) is the prob-
ability measure on R3 with pt(x,R3) = 1, and (1) follows. (2) is trivial. From the
semigroup property e−sLpe−tLp1lA = e−(s+t)Lp1lA, the Chapman-Kolmogorov equal-
ity (7.102) follows. Hence pt(x, A) is a probability transition kernel. We write
E for EQx for notational simplicity. From the identity E[1lA(Yt)E[f(Yr)|σ(Yt)]] =
E[1lA(Yt)f(Yr)] for r > t, it follows that∫1lA(y)E[f(Yr)|Yt = y]Pt(dy) =
∫Pt(dy)1lA(y)
∫f(y′)p(r − t, y, dy′),
where Pt(dy) denotes the distribution of Yt on R3. Thus
E[f(Yr)|Yt = y] =
∫f(y′)p(r − t, y, dy′)
follows almost everywhere y with respect to Pt(dy). Then
E[f(Yr)|σ(Yt)] =
∫f(y)p(r − t, Yt, dy)
and
E[1lA(Yr)|σ(Yt)] = p(r − t, Yt, A) (7.103)
follow. By using (7.103) and the Chapman-Kolmogorov equality we can show that
E
[1lA(Yt+s)
n∏j=0
1lAj(Ytj)
]= E
[E[1lA(Yt)|σ(Ys)
] n∏j=0
1lAj(Ytj)
]
for t0 ≤ · · · ≤ tn ≤ s. This implies that E[1lA(Yt+s)|Mt] = E[1lA(Yt)|σ(Ys)]. Then Y
is Markov with respect to the natural filtration under the measure Qx. 2
(Step 4) We extend Y = (Yt)t≥0 to a process on the hole real line R. Set X+ =
X+×X+, M = B(X+)×B(X+) and Qx = Qx×Qx. Let (Xt)t∈R be the stochastic
process on the product space (X+, M , Qx), defined by for ω = (ω1, ω2) ∈ X+,
Xt(ω) =
Yt(ω1), t ≥ 0,
Y−t(ω2), t < 0.(7.104)
180
Note that X0 = x almost surely with respect to Qx and Xt is continuous in t almost
surely. It is trivial to see that Xt, t ≥ 0, and Xs, s ≤ 0, are independent, and
Xtd= X−t.
(Step 5) Proof of Proposition 7.28:
The image measure of Qx on (X ,B(X )) with respect to X is denoted by P x, i.e.,
P x = Qx X−1. Let Xt(ω) = ω(t), t ∈ R, ω ∈X , be the coordinate process. Then
we can see that
Xtd= Yt (t ≥ 0), Xt
d= Y−t (t ≤ 0). (7.105)
Since by (Step 3), (Yt)t≥0 and (Y−t)t≤0 are Markov processes with respect to the
natural filtration σ(Ys, 0 ≤ s ≤ t) and σ(Ys,−t ≤ s ≤ 0), respectively, (Xt)t≥0 and
(Xt)t≤0 are also Markov processes with respect to (F +t )t≥0 and (F−
t )t≤0, respec-
tively, where
F +t = σ(Xs, 0 ≤ s ≤ t), F−
t = σ(Xs,−t ≤ s ≤ 0).
Thus the diffusion property (3) follows. We also see that (Xs)s≤0 and (Xt)t≥0 are
independent and X−td= Xt by (7.105) and (Step 4). Thus reflection symmetry (2)
follows.
Lemma 7.31 Let −∞ < t0 ≤ t1 ≤ · · · ≤ tn. Then∫dµpEP x [f0(Xt0) · · · fn(Xtn)] = (f0, e
−(t1−t0)Lpf1 · · · e−(tn−tn−1)Lpfn)Hp . (7.106)
Proof: Let t0 ≤ · · · ≤ tn ≤ 0 ≤ tn+1 ≤ · · · tn+m. Then we have by the independence
of (Xs)s≤0 and (Xt)t≥0,∫dµpEP x
[f0(Xt0) · · · fn+m(Xtn+m)
]=
∫dµpEP x [f0(Xt0) · · · fn(Xtn)]EP x
[fn+1(Xtn+1) · · · fn+m(Xtn+m)
].
Since we have
EP x
[fn+1(Xtn+1) · · · fn+m(Xtn+m)
]=(e−tn+1Lpfn+1e
−(tn+2−tn+1)Lpfn+2 · · · e−(tn+m−tn+m−1)Lpfn+m
)(x) (7.107)
181
and
EP x [f0(Xt0) · · · fn(Xtn)] = EP x
[f0(Y−t0) · · · fn(Y−tn)
]=
(e+tnLpfne
−(tn−tn−1)Lpfn−1 · · · e−(t1−t0)Lpf1
)(x).(7.108)
By (7.107) and (7.108) we have∫dµpEP x
[f0(Xt0) · · · fn+m(Xtn+m)
]= (e+tnLpfn · · · e−(t1−t0)Lpf1, e
−tn+1Lpfn+1 · · · e−(tn+m−tn+m−1)Lpfn+m)Hp
= (f1, e−(t1−t0)Lpf2 · · · e−(tn+m−tn+m−1)Lpfn+m)Hp .
Hence (7.106) follows. 2
From Lemma 7.31 it follows that for any s ∈ R,∫dµpEP x
[n∏j=0
fj(Xtj)
]=
∫dµpEP x
[n∏j=0
fj(Xtj+s)
].
Hence shift invariance (4) is obtained. 2
We denote Ex for EP x in what follows.
7.4 Absence of ground state
7.4.1 The Nelson model by path measures
Now we construct a Feynman-Kac formula for e−tH by using the diffusion process
X. Let φE(f) be the Euclidean scalar field on a probability space (QE,ΣE, µE),
which is the Gaussian random variable indexed by f ∈ L2(R4) with EµE[φE(f)] = 0
and the covariance given by
EµE[φE(f)φE(g)] =
1
2(f , g).
Euclidean scalar field L2(QE) and L2(Q) are connected through some isometry jt.
Let jt : L2(R3)→ L2(R4) be given by
jtf(x0, x) =1
2π
∫dk0e
−i(t−x0)k0(ω1/2(ω2 + |k0|2)−1/2f
)(x). (7.109)
182
Then we can have the formula (jsf, jtg) = (f, e−|t−s|ωg). In particular
j∗t js = e−|t−s|ω. (7.110)
Let Jt = Γ(jt) : L2(Q)→ L2(QE) be the isometry defined by Jt1l = 1lE and
Jt :n∏j=1
φ(fj):=:n∏j=1
φE(jtfj): .
From the definition of Jt, the identity
J∗tJs = e−|t−s|Hf (7.111)
follows. Thus the semigroup e−tHf can be factorized by Jt and it can be expressed
as
(Φ, e−tHf Ψ)L2(Q) = (J0Φ, JtΨ)L2(QE). (7.112)
Theorem 7.32 (Feynman-Kac formula)
Suppose Assumptions 7.8, 7.10, 7.12, 7.16 and 7.18. Then we have
(F, e−THG)H =
∫dµpEx
[(J0F (X0), eφE(KT )JTG(XT ))L2(QE)
], (7.113)
where KT =∫ T
0jsω−1/2ρ(· − Xs)ds is an L2(R4)-valued integral. In particular it
follows that
(1l, e−TH1l)H =
∫dµpEx
[e(1/2)
∫ T0 dt
∫ T0 dsW (Xt,Xs,|t−s|)
], (7.114)
where 1l ∈ L2(R3 ×Q) and
W (X, Y, |t|) =1
2(ρ(· −X), ω−1e−|t|ωρ(· − Y )). (7.115)
Proof: By the Trotter-Kato product formula:
e−tH = s-limn→∞
(e−(t/n)Lpe−(t/n)φρe−(t/n)Hf
)n,
183
the factorization formula (7.111), and Markov property of Et = JtJ∗t , we have
(F, e−tHG) = limn→∞
∫dµpEx
[(J0F (X0), e
−∑nj=0
tnφE(j tj
nρ(·−Xtj/n))
JtG(Xt)
)].
(7.116)
Note that the map R → L2(R3), s 7→ ω−1/2ρ(· −Xs), is strongly continuous almost
surely. Hence the map R→ L2(QE), s 7→ φE(jsρ(·−Xs)), is also strongly continuous.
By a simple limiting argument (7.113) follows. Let F = G = 1l. Since φE is a
Gaussian random variable, we have
(1l, e−tH1l) =
∫dµpEx
[(1l, eφE(KT )1l)
]=
∫dµpEx
[e
(1/4)‖KT ‖2L2(R4)
].
Hence ‖KT‖2L2(R4) =
∫ T0dt∫ T
0dsW (Xt, Xs, |t− s|) and (7.114) follows. 2
7.4.2 Absence of ground states
From Theorem 7.32 we can obtain a useful lemma to show the absence of ground
states.
Theorem 7.33 (Positivity improving) Suppose Assumptions 7.8, 7.10, 7.12,
7.16 and 7.18. Then e−tH is positivity improving for all t > 0.
Proof: Let F,G ∈ L2(R3 × Q) be such that F ≥ 0 and G ≥ 0 but F 6≡ 0 and
G 6≡ 0. Define DF = x ∈ R3|F (x, ·) 6≡ 0 and DG = x ∈ R3|G(x, ·) 6≡ 0. Note
that∫
DFdx > 0 and
∫DGdx > 0. Let K x = ω ∈ X |X0(ω) = x,Xt(ω) ∈ DF. It
follows that∫DF
dµp
∫K x
dP x = (1lDG , e−tLp1lDF )Hp = (ϕp1lDG , e
−tKϕp1lDF )L2et infσ(K) > 0
by Lemma 2.49. Thus R3 ×Q ⊃ ∪x∈DFK x has a positive measure with respect to
dµpdPx and
(F, e−tHG) ≥∫
DF
dµp
∫K x
dP x(J0F, eφ(Kt)JtG(Xt)) > 0,
since (J0F (X0(ω)), eφ(Kt(ω))JtG(Xt(ω))) > 0 for ω ∈ ∪x∈DFK x. Then the theorem
follows. 2
By Theorem 7.33 and Perron-Frobenius arguments, we immediately have the
corollary.
184
Corollary 7.34 (Uniqueness of ground state) Suppose Assumptions 7.8, 7.10,
7.12, 7.16 and 7.18. Then the ground state of ϕg is unique and ϕg > 0 if it exists.
In particular (1l, ϕg) > 0.
From Corollary 7.34 we can see that e−TH1l/‖e−TH1l‖ converges to the ground state
as T →∞ if the ground state exists. We then define γ(T ) by
γ(T ) =(1l, e−TH1l)2
(1l, e−2TH1l), T > 0. (7.117)
Lemma 7.35 Suppose Assumptions 7.8, 7.10, 7.12, 7.16 and 7.18. Let P∆, ∆ ⊂ R,
denote the spectral projection of H associated with ∆ ∩ σ(H). Let E = infσ(H).
Then it follows that limT→∞ γ(T ) = ‖PE1l‖2. In particular H has a ground state
if and only if limT→∞ γ(T ) 6= 0.
Proof: Assume that E = 0. Thus limT→∞ e−TH = PE. If 0 is an eigenvalue, then
by Corollary 7.34 and Perron-Frobenius arguments, PE = (u, ·)u for some u > 0. It
follows that limT→∞ γ(T ) = (u, 1l)2 > 0. Next we prove the sufficient part. Assume
now that there exists a sequence Tn → +∞ such that δ(Tn) ≥ ε2 > 0. This implies
that (1l, e−TnH1l) ≥ ε(1l, e−2TnH1l)1/2. Letting n → ∞, we obtain that ‖PE1l‖ ≥ ε.
Then H has a ground state. 2
The denominator of γ(T ) is computed as
‖e−TH1l‖2 = (1l, e−2TH1l) =
∫dµpEx
[e(1/2)
∫ T−T ds
∫ T−T dtW (Xs,Xt,|s−t|)
]by the shift invariance (Proposition 7.28) of Xt. Then γ(T ) can be expressed as
γ(T ) =
(∫dµpEx
[e(1/2)
∫ T0 ds
∫ T0 dtW (Xs,Xt,|s−t|)
])2
∫dµpEx
[e(1/2)
∫ T−T ds
∫ T−T dtW (Xs,Xt,|s−t|)
] . (7.118)
Let µT be the probability measure on (R3 × X ,B(R3) × B(X )) defined by for
O ∈ B(R3)×B(X ),
µT (O) =1
ZT
∫dµpEx
[1lOe
(1/2)∫ T−T ds
∫ T−T dtW (Xs,Xt,|s−t|)
], (7.119)
where ZT denotes the normalizing constant such that µT becomes a probability
measure.
185
Figure 13:
∫ 0
−T
∫ 0
−T+
∫ T
0
∫ T
0
=
∫ T
−T
∫ T
−T−2
∫ 0
−T
∫ T
0
Lemma 7.36 Suppose Assumptions 7.8, 7.10, 7.12, 7.16 and 7.18. Then it follows
that
γ(T ) ≤ EµT[e−
∫ 0−T ds
∫ T0 dtW (Xs,Xt,|s−t|)
]. (7.120)
Proof: The numerator of (7.118) can be estimated by the Schwartz inequality with
respect to dµp and the reflection symmetry of X, and then(∫dµpEx
[e(1/2)
∫ T0 ds
∫ T0 dtW
])2
≤∫dµp
(Ex[e(1/2)
∫ T0 ds
∫ T0 dtW
])(Ex[e(1/2)
∫ 0−T ds
∫ 0−T dtW
]).
Since Xt and Xs for s ≤ 0 ≤ t are independent, thus we have(∫dµpEx
[e(1/2)
∫ T0 ds
∫ T0 dtW
])2
≤∫dµpEx
[e(1/2)(
∫ T0 ds
∫ T0 dtW +
∫ 0−T ds
∫ 0−T dtW )
].
Moreover∫ 0
−T
∫ 0
−T +∫ T
0
∫ T0
=∫ T−T
∫ T−T −2
∫ 0
−T
∫ T0
yields that (Figure 13)(∫dµpEx
[e(1/2)
∫ T0 ds
∫ T0 dtW
])2
≤∫dµpEx
[e−
∫ 0−T ds
∫ T0 dtW +(1/2)
∫ T−T ds
∫ T−T dtW
].
Then the lemma follows. 2
In order to show that the right-hand side of (7.120) converges to zero as T →∞,
we estimate its upper bound. Let
W∞(X, Y, |t|) =1
2
∫dkρ(k)2e−ik·(X−Y )
|k|e−|t||k| (7.121)
186
or it is expressed in the position representation as
W∞(X, Y, |t|) =1
2(ρ(· −X), ω−1
∞ e−|t|ω∞ρ(· − Y )) (7.122)
with
ω∞ =√−∆. (7.123)
The next proposition on the upper and lower Gaussian bound of the integral kernel
e−tω2(x, y) is the key ingredient of the proof of the absence of ground states of H.
Proposition 7.37 Suppose Assumption 7.12. Then the semigroup e−tω2
has an
integral kernel e−tω2(x, y), and there exist constants C1, · · · , C4 such that
C1e−C2tω2
∞(x, y) ≤ e−tω2
(x, y) ≤ C3e−C4tω2
∞(x, y) (7.124)
for t ≥ 0 and a.e. x, y ∈ R3.
Proof: Conjugating by the unitary U : L2(R3) → L2(R3, c2(x)dx), f 7→ c−1f , we
obtain
ω2 = Uω2U−1 = h0 +m2(x),
where
h0 = h0(D, x) = c−2(x)3∑
µν=1
Dµaµν(x)Dν .
Throughout (f, g)2 denotes∫f(x)g(x)c2(x)dx and ‖f‖pp =
∫|f(x)|pc2(x)dx. Note
that C0
∫|f(x)|pdx ≤ ‖f‖pp ≤ C1
∫|f(x)|pdx. Let f ∈ C∞0 (R3). Then
C0(f, h0f)2 ≤ E1l(f, f) ≤ C1(f, h0f)2.
From this D(h1/20 ) = H1(R3) follows and
C0(h1/20 f, h
1/20 f)2 ≤ E1l(f, f) ≤ C1(h
1/20 f, h
1/20 f)2 (7.125)
for f ∈ H1(R3). Notice that e−tω(x, y) denotes an integral kernel of e−tω with respect
to the measure c2(x)dx, while e−tω(x, y) is that of e−tω with respect to dx. Since
(f, e−tω2
g)L2(R3) =
∫dx
∫dyf(x)e−tω
2
(x, y)g(y) = (Uf, e−tω2
Ug)2
=
∫c2(x)dx
∫c2(y)dy
1
c(x)f(x)e−tω
2
(x, y)1
c(y)g(y),
187
we note that
e−tω2
(x, y) = c(x)e−tω2
(x, y)c(y)
almost everywhere. So it suffices to prove proposition for e−tω2. We know from
[PE84, Theorems 3.4 and 3.6] that e−th0 has an integral kernel with
C1e−C2tω∞(x, y) ≤ e−th0(x, y) ≤ C3e
−C4tω∞(x, y) (7.126)
for a.e. x, y ∈ R3 and all t > 0. Notice that e−t(h0+V ), supx |V (x)| <∞, is positivity
preserving and bounded on L∞(R3). This can be proven by the Trotter product
formula. Then by (7.126) we see that ‖e−th0f‖∞ ≤ Ct−3/4‖f‖2, which is equivalent
to
‖f‖26 ≤ C(h
1/20 f, h
1/20 f)2 (7.127)
by Proposition 7.5. Since m2(x) ≥ 0, the upper bound follows from the Trotter
product formula. Let us now prove the lower bound, following [Sem97, Theorem
6.1]. Since m2(x) ≤ 〈x〉−β with β > 2, we see that m2 ∈ L3/2(R3). Then we have
(m2f, f) ≤ ‖m2‖3/2‖f‖26. By the Sobolev inequality we see that ‖f‖2
6 ≤ C1E1l(f, f),
and then together with (7.125),
γ(m2f, f)2 ≤ (h1/20 f, h
1/20 f)2
for some γ > 0. Set now w(x) = −γm2(x)/4. Then h0 + 2w ≥ 12h0 in the sense of
form. We see, together with (7.127), that
‖f‖26 ≤ C2((h0 + 2w)1/2f, (h0 + 2w)1/2f)2. (7.128)
By Proposition 7.5 and the fact that e−t(h0+w) is positivity preserving and bounded
on L∞(R3), (7.128) is equivalent to
‖e−t(h0+2w)f‖∞ ≤ C3t−3/4‖f‖2
and e−t(h0+2w) is ultracontractive. Then e−t(h0+2w) has an integral kernel, further-
more it can be estimated as
e−t(h0+2w)(x, y) ≤ C4t−3/2
almost everywhere. We prove in Lemma 7.38 that λ 7→ e−t(h0+λw)(x, y) is logarith-
mically convex for all t > 0 and a.e. x, y ∈ R3. Then we have
e−t(h0+w)(x, y) = e−t(h0+ 12
0w+ 12
2w)(x, y) ≤ t−3/4e−th0(x, y)1/2. (7.129)
188
Applying again the log-convexity we get that
e−th0(x, y) = e−t(h0+sm2+(1−s)w)(x, y) ≤ e−t(h0+m2)(x, y)se−t(h0+w)(x, y)1−s
with s = γ/(4 + γ). Hence using (7.129) we obtain
e−th0(x, y)(1+s)/2t(1−s)3/4 ≤ e−t(h0+m2)(x, y)s,
which, together with (7.126), implies the proposition. 2
It remains to show the log-convexity of e−t(h0+w)(x, y).
Lemma 7.38 R 3 λ 7→ e−t(h0+λw)(x, y) ∈ R is logarithmically convex for all t > 0
and a.e. x, y ∈ R3, i.e., for 0 ≤ s ≤ 1,
e−t(h0+(sλ+(1−s)λ′)w)(x, y) ≤ e−t(h0+λw)(x, y)se−t(h0+λ′w)(x, y)1−s.
Proof: Set t = 1. By the Trotter product formula we have
e−(h0+λw)(x, y) =(s− lim
n→∞
(e−h0/ne−λw/n
)n)(x, y). (7.130)
Let Aλ(x, y) and Bλ(x, y) be the kernels of two operators Aλ and Bλ assumed to be
log-convex in λ. Then the kernel of AλBλ:
AλBλ(x, y) =
∫R3
Aλ(x, z)Bλ(z, y)dz
is also log-convex in λ. Then the kernel of e−h0/ne−λw/n(x, y) = e−h0/n(x, y)e−λw(y)/n
is log-convex in λ. Then the lemma follows from the Trotter product formula (7.130).
2
Corollary 7.39 (Positivity improving) Suppose Assumption 7.12. Then e−tω2
is positivity improving.
Proof: This immediately follows from the Gaussian bound (7.124). 2
Corollary 7.40 Suppose Assumption 7.12. Then it follows that
‖ω−n/2f‖ ≤ C‖ω−n/2∞ f‖.
189
Proof: Since ω−n/2 = Cn∫∞
0e−tω
2t(n+4)/4dt with Cn = (
∫∞0e−ss(n+4)/4ds)−1. Hence
the corollary follows from Proposition 7.37. 2
Lemma 7.41 Suppose Assumptions 7.12 and 7.16. Then
W (X, Y, |t|) ≥ 0, W∞(X, Y, |t|) ≥ 0
and there exist constants Cj > 0, j = 1, 2, 3, 4, such that
C1W∞(X, Y,C2|t|) ≤ W (X, Y, |t|) ≤ C3W∞(X, Y,C4|t|) (7.131)
for all X, Y ∈ R3 and t ∈ R. In particular it follows that
γ(T ) ≤ EµT[e−C1
∫ 0−T ds
∫ T0 dtW∞(Xs,Xt,C2|s−t|)
]. (7.132)
Proof: Set ρX(x) = ρ(x −X). We note that the function f(x) = e−√x on [0,∞) is
completely monotone, i.e., (−1)ndf(x)/dxn ≥ 0 and that f(+0) = 0. Then there
exists a Borel probability measure m on [0,∞) such that
e−√x =
∫ ∞0
e−sxdm(s)
and it is indeed exactly given by
dm(s) =1
2√π
e−1/(4s)
s3/2ds.
Hence
e−tω =
∫ ∞0
e−st2ω2
dm(s) =1
2√π
∫ ∞0
te−t2/(4s)
s3/2e−sω
2
ds.
It follows that
W (X, Y, |t|) =1
2
∫ ∞t
dr(ρX , e−rωρY ) =
1
4√π
∫ ∞t
dr
∫ ∞0
re−r2/(4p)
p3/2(ρX , e
−pω2
ρY )dp.
Hence W (X, Y, |t|) > 0 follows, since e−pω2
is positivity improving for p > 0.
W∞(X, Y, |t|) > 0 also follows in the same way as above. Since ρX and ρY are
190
nonnegative, by the Gaussian bound c1e−c2tω∞(x, y) ≤ e−tω
2(x, y) ≤ c3e
−c4tω∞(x, y),
we can see that by changing a variable,
c1c2W∞(X, Y,√c2|t|) ≤ W (X, Y, |t|) ≤ c3c4W∞(X, Y,
√c4|t|).
Then the lemma follows. 2
Let us take λ such that1
δ + 1< λ < 1, (7.133)
where δ is the positive constant given in Assumption 7.10. Let
AT = R3 ×
sup|s|≤T|Xs| ≤ T λ
⊂ R3 ×X . (7.134)
We divide the right-hand side of (7.132) into EµT [1lAT · · · ]+EµT [1lAcT · · · ]. Then in or-
der to prove the absence of ground state it is enough to show that limT→∞ EµT [1lAT · · ·] =
0 and limT→∞ EµT[1lAcT · · ·
]= 0.
Lemma 7.42 [LMS02] Suppose Assumptions 7.8, 7.10, 7.12, 7.16 and 7.18. Then
it follows that
limT→∞
EµT[1lAT e
−C1
∫ 0−T ds
∫ T0 dtW∞(Xs,Xt,C2|s−t|)
]= 0. (7.135)
Proof: Since the integral kernel of e−|t|ω∞ is
e−|t|ω∞(x, y) =1
π2
|t|(|x− y|2 + |t|2)2
,
we have
W∞(X, Y, |t|) =1
4π2
∫dx
∫dy
ρ(x)ρ(y)
|(x−X)− (y − Y )|2 + |t|2. (7.136)
On AT we know that |(Xs − x)− (Xt − y)|2 + |t− s|2 ≤ 8T 2λ + 2|x− y|2 + |t− s|2.
Let
∆T = (s, t)|0 ≤ s ≤ T, 0 ≤ t ≤ T, 0 ≤ s+ t ≤ T/√
2,∆′T = (s, t)|0 ≤ s ≤ T/
√2,−s ≤ t ≤ s.
191
Since ∫ 0
−Tdt
∫ T
0
dt1
a2 + |t− s|2≥∫ ∫
∆T
dsdt1
a2 + |s+ t|2
=
∫ ∫∆′T
dsdt1
a2 + s2= log
(a2 + T 2/2
a2
),
we have
1lAT
∫ 0
−Tds
∫ T
0
dtW∞(Xs, Xt, C2|s− t|)
≥ 1
4π21lAT
∫ 0
−Tds
∫ T
0
dt
∫dxdy
ρ(x)ρ(y)
8T 2λ + 2|x− y|2 + C2|t− s|2
≥ 1
4C2π21lAT
∫dxdyρ(x)ρ(y) log
(8T 2λ + 2|x− y|2 + C2T
2/2
8T 2λ + 2|x− y|2
).
Note that ρ ≥ 0 and λ < 1. Since the right-hand side above goes to +∞ as T →∞,
(7.135) follows.
Lemma 7.43 Suppose Assumptions 7.8, 7.10, 7.12, 7.16 and 7.18. Then it follows
that
limT→∞
EµT[1lAcT e
−C1
∫ 0−T ds
∫ T0 dtW∞(Xs,Xt,C2|s−t|)
]= 0. (7.137)
Proof: Note that
∫ 0
−Tds
∫ T
0
dtW∞(Xs, Xt, |s− t|) ≤T
2‖ω−1∞ ρ‖2 (7.138)
and ∫ T
−Tds
∫ T
−TdtW∞(Xs, Xt, |s− t|) ≤ 4T‖ω−1
∞ ρ‖2. (7.139)
Then
EµT[1lAcT e
−∫ 0−T ds
∫ T0 dtW∞
]≤ e(T/2)‖ω−1
∞ ρ‖2EµT[1lAcT
].
192
By the Schwartz inequality we have
e(T/2)‖ω−1∞ ρ‖2EµT
[1lAcT
]= e(T/2)‖ω−1
∞ ρ‖2
∫dµpEx
[1lAcT e
(1/2)∫ T−T ds
∫ T−T dtW
]∫dµpEx
[e(1/2)
∫ T−T ds
∫ T−T dtW
]≤ e(T/2)‖ω−1
∞ ρ‖2
(∫dµpEx
[e∫ T−T ds
∫ T−T dtW
])1/2
∫dµpEx
[e(1/2)
∫ T−T ds
∫ T−T dtW
] ∫ dµpEx[1lAcT
]. (7.140)
By Lemma 7.41 bounds
C1
∫ T
−Tds
∫ T
−TdtW∞(Xs, Xt, C2|s− t|) ≤
∫ T
−Tds
∫ T
−TdtW (Xs, Xt, |s− t|)
and ∫ T
−Tds
∫ T
−TdtW (Xs, Xt, |s− t|) ≤ C3
∫ T
−Tds
∫ T
−TdtW∞(Xs, Xt, C4|s− t|)
are derived. Then we obtain(∫dµpEx
[e∫ T−T ds
∫ T−T dtW
])1/2
∫dµpEx
[e(1/2)
∫ T−T ds
∫ T−T dtW
] ≤(∫dµpEx
[eC3
∫ T−T ds
∫ T−T dtW∞(Xs,Xt,C4|s−t|)
])1/2
∫dµpEx
[e(C1/2)
∫ T−T ds
∫ T−T dtW∞(Xs,Xt,C2|s−t|)
]and by (7.139) there exists ε > 0 such that(∫
dµpEx[e∫ T−T ds
∫ T−T dtW
])1/2
∫dµpEx
[e(1/2)
∫ T−T ds
∫ T−T dtW
] ≤ eεT‖ω−1∞ ρ‖2 . (7.141)
It remains to estimate∫dµpEx
[1lAcT
]in (7.140). There exists an at most polynomially
growth function ξ(T ) such that∫dµpEx
[1lAcT
]≤ ξ(T ) exp
(−cT λ(δ+1)
)(7.142)
with some constant c > 0. This is proven in Lemma 7.46 below. By (7.140), (7.141)
and (7.142) we have
limT→∞
EµT [1lAcT ] ≤ limT→∞
ξ(T )e−cTλ(δ+1)
e(ε+1/2)T‖ω−1∞ ρ‖2 = 0, (7.143)
193
since 1δ+1
< λ < 1. Then (7.137) follows. 2
Let us now consider some path properties of X to show (7.142).
Proposition 7.44 Let P (B) =∫
1lBdµpdPx be the probability measure on R3 ×X
and Λ > 0. Suppose Assumptions 7.8, 7.10, and 7.18. Suppose that f ∈ C(R3) ∩D(L
1/2p ). Then it follows that
P
(sup
0≤s≤T|f(Xs)| ≥ Λ
)≤ e
Λ
[(f, f)Hp + T (L1/2
p f, L1/2p f)Hp
]1/2. (7.144)
Proof: The proof is a modification of that of [KV86, Lemma 1.4 and Theorem 1.12].
Set Tj = Tj/2n, j = 0, 1, ..., 2n and we fix T and n. Let G = x ∈ R3||f(x)| ≥ Λ,then the stopping time τ is defined by
τ = infTj ≥ 0|XTj ∈ G.
Then it follows that
P
(sup
j=0,...,2n|f(XTj)| ≥ Λ
)= P (τ ≤ T ).
We estimate the right-hand side above. Let 0 < χ < 1 be fixed and we choose a
suitable χ later. We see that
P (τ ≤ T ) =
∫dµpEx[1lτ≤T ] ≤
∫dµpEx
[χτ−T
]≤ χ−T
∫dµpEx[χτ ] ≤ χ−T
(∫dµp(Ex[χτ ])2
)1/2
. (7.145)
Let 0 ≤ ψ be any function such that ψ(x) ≥ 1 on G. Then the Dirichlet principle∫dµp(Ex[χτ ])2 ≤ (ψ, ψ)Hp +
χT/2n
1− χT/2n(ψ, (1l− e−(T/2n)Lp)ψ)Hp (7.146)
follows. We prove this in the next lemma. Inserting
|f(x)|/Λ =
≥ 1, x ∈ G,|f(x)|/Λ, x ∈ Gc,
194
into ψ in (7.146), we have∫dµp(Ex[χτ ])2 ≤ 1
Λ2(f, f)Hp +
χT/2n
1− χT/2n1
Λ2(|f |, (1l− e−(T/2n)Lp)|f |)Hp . (7.147)
Since e−(T/2n)Lp is positivity improving and then
(|f |, (1l− e−(T/2n)Lp)|f |)Hp ≤ (f, (1l− e−(T/2n)Lp)f)Hp ,
we have by (7.145),
P
(sup
j=0,...,2n|f(XTj)| ≥ Λ
)≤ χ−T
Λ
[(f, f)Hp +
χT/2n
1− χT/2n(f, (1l− e−(T/2n)Lp)f)Hp
]1/2.
Set χ = e−1/T . Then by χT/2n
1−χT/2n ≤ 2n, we have
P
(sup
j=0,...,2n|f(XTj)| ≥ Λ
)≤ e
Λ
[(f, f)Hp + 2n(f, (1l− e−(T/2n)Lp)f)Hp
]1/2. (7.148)
Since (f, (1l− e−(T/2n)Lp)f)Hp ≤ (T/2n)(L1/2p f, L
1/2p f), we obtain that
P
(sup
j=0,...,2n|f(XTj)| ≥ Λ
)≤ e
Λ
[(f, f)Hp + T (L1/2
p f, L1/2p f)Hp
]1/2. (7.149)
Take n → ∞ on both sides of (7.149). By the Lebesgue dominated convergence
theorem,
limn→∞
P
(sup
j=0,...,2n|f(XTj)| ≥ Λ
)= P
(limn→∞
supj=0,...,2n
|f(XTj)| ≥ Λ
).
Since f(Xt) is continuous in t, limn→∞ supj=0,...,2n |f(XTj)| = sup0≤s≤T |f(Xs)| fol-
lows. Then we complete the proposition. 2
It remains to show the Dirichlet principle (7.146).
Lemma 7.45 (Dirichlet principle) Suppose Assumptions 7.8, 7.10 and 7.18.
Then it follows that∫dµp(Ex[χτ ])2 ≤ (ψ, ψ)Hp +
χT/2n
1− χT/2n(ψ, (1l− e−(T/2n)Lp)ψ)Hp (7.150)
for any function ψ ≥ 0 such that ψ(x) ≥ 1 on G.
195
Proof: Define the function ψχ by ψχ(x) = Ex[χτ ]. By the definition of τ we can see
that
ψχ(x) = 1, x ∈ G, (7.151)
since τ = 0 when Xs stars from the inside of G. Let Ft = σ(Xs, 0 ≤ s ≤ t) be the
natural filtration of (Xt)t≥0. By the Markov property of X, we can directly see that
e−(T/2n)Lpψχ(x) = Ex[EXT/2n [χτ ]] = Ex[Ex[χτθT/2n |FT/2n ]] = Ex[χτθT/2n ], (7.152)
where θt is the shift on X , which is defined by (θtω)(s) = ω(s+ t) for ω ∈X . Note
that
(τ θT/2n)(ω) = τ(ω)− T/2n ≥ 0, (7.153)
when x = X0(ω) ∈ Gc. Hence by (7.152) and (7.153) we have the identity:
χT/2n
e−(T/2n)Lpψχ(x) = ψχ(x), x ∈ Gc. (7.154)
It is trivial to see that∫dµp(Ex[χτ ])2 = (ψχ, ψχ)Hp ≤ (ψχ, ψχ)Hp +
χT/2n
1− χT/2n(ψχ, (1l− e−(T/2n)Lp)ψχ)Hp .
Let us define (f, g)G =∫Gdµpf(x)g(x). By the relation (7.154) we can compute the
right-hand side above as
(ψχ, ψχ)G +χT/2
n
1− χT/2n(ψχ, (1l− e−(T/2n)Lp)ψχ)G. (7.155)
Since
(ψχ, (1l− e−(T/2n)Lp)ψχ)G
= (ψχ1lG, (1l− e−(T/2n)Lp)ψχ1lG)Hp + (ψχ1lG, (1l− e−(T/2n)Lp)ψχ1lGc)Hp
= (ψχ1lG, (1l− e−(T/2n)Lp)ψχ1lG)Hp − (ψχ1lG, e−(T/2n)Lpψχ1lGc)Hp
≤ (ψχ1lG, (1l− e−(T/2n)Lp)ψχ1lG)Hp .
Hence∫dµp(Ex[χτ ])2 ≤ (ψχ1lG, ψχ1lG)Hp +
χT/2n
1− χT/2n(ψχ1lG, (1l− e−(T/2n)Lp)ψχ1lG)Hp .
(7.156)
196
Note that ψχ1lG(x) ≤ ψ(x) for all x ∈ R3. Then
(ψχ1lG, ψχ1lG)Hp +χT/2
n
1− χT/2n(ψχ1lG, (1l− e−(T/2n)Lp)ψχ1lG)Hp
≤ (ψ, ψ)Hp +χT/2
n
1− χT/2n(ψ, (1l− e−(T/2n)Lp)ψ)Hp . (7.157)
Combining (7.156) with (7.157), we prove the lemma. 2
Lemma 7.46 (7.142) holds.
Proof: Suppose that f ∈ C∞(R3), f(−x) = f(x) and
f(x) =
|x|, |x| ≥ T λ,≤ |x|, T λ − 1 < |x| < T λ,0, |x| ≤ T λ − 1.
Then we see that∫dµpEx
[1lAcT
]=
∫dµpEx
[1lsup|s|<T |Xs|>Tλ
]=
∫dµpEx
[1lsup|s|<T |f(Xs)|>Tλ
]. (7.158)
By the reflection symmetry of (Xt)t∈R we have∫dµpEx
[1lsup|s|<T |f(Xs)|>Tλ
]= 2
∫dµpEx
[1lsup0≤s≤T |f(Xs)|>Tλ
]and by Proposition 7.44 we have∫
dµpEx
[sup|s|<T|f(Xs)| > T λ
]≤ 2e
T λ
[(f, f)Hp + T (L1/2
p f, L1/2p f)Hp
]1/2. (7.159)
We estimate the right-hand side of (7.159). First we show fϕg ∈ D(K). Let
C∞0 (R3) 3 fR(x) = X (x/R)f(x), where X ∈ C∞0 (R3) and
X (x) =
1, |x| < 1,< 1, 1 ≤ |x| ≤ 2,0, |x| > 2.
197
Since Aµν satisfies the Lipshitz condition (Assumption 7.18), Aµν ∈ W 1,∞(R3). Then
by Lemma 7.9, we see that ϕp ∈ H2(R3), fRϕp ∈ D(K), (K − E)ϕp = 0, and
KfRϕp =3∑
µ,ν=1
(DµAµν(DνfR)ϕp + (Dµf)AµνDνϕp) + EfRϕp.
We see that
fRϕp → fϕp,
(DµfR)AµνDνϕp → (Dµf)AµνDνϕp,
DµAµν(DνfR)ϕp = DµAµν · (DνfR)ϕp + Aµν ·Dµ(DνfR) · ϕp + Aµν(DνfR) ·Dµϕp
→ ((DµAµν)(Dνf) + Aµν(DµDνf))ϕp
as R → ∞ in L2(R3). Since K is closed, fϕp ∈ D(K) follows. By the estimate
above we also see that
(L1/2p f, L1/2
p f)Hp = (fϕp, (Dµf)AµνDνϕp + (DνAµν)(Dνf)ϕp + Aµνfνµϕp) .
By the spatial super-exponential decay ϕp(x) ≤ Ce−γ|x|δ+1
derived in (7.87) we have
‖fϕp‖2 =
∫f(x)2ϕ2
p(x)dx ≤ C2e−2γTλ(δ+1)
∫|x|2e−2γ|x|δ+1
dx. (7.160)
Note that Dµf,DµDνf ∈ L∞(R3). Then
(L1/2p f, L1/2
p f)Hp ≤ ‖fϕp‖‖AµνDνϕp + Aνµνϕp + Aµνϕp‖
≤ C ′e−γTλ(δ+1)‖AµνDνϕp + Aνµνϕp + Aµνϕp‖
follows. Similarly
(f, f)Hp ≤ C2e−2γTλ(δ+1)
∫|x|2e−2γ|x|δ+1
dx
is also derived. Hence we have
EµT[1lAcT
]≤ T−λ
√a+ Tbe−(γ/2)Tλ(δ+1)
with some constant a and b. This completes the proof. 2
Now we are in the position to state the main theorem.
198
Theorem 7.47 (Absence of ground state) Suppose Assumptions 7.8, 7.10, 7.12,
7.16 and 7.18. Then there is no ground states of H.
Proof: Since γ(T ) ≤ EµT[e−C1
∫ 0−T ds
∫ T0 dtW∞(Xs,Xt,C2|s−t|)
]and
limT→∞
EµT[e−C1
∫ 0−T ds
∫ T0 dtW∞(Xs,Xt,C2|s−t|)
]= 0
by Lemmas 7.42 and 7.43, we obtain limT→∞ γ(T ) = 0. Then the theorem follows.
2
Acknowledgments
FH acknowledges support of Grant-in-Aid for Science Research (B) 20340032 from
JSPS and Grant-in-Aid for Challenging Exploratory Research 22654018 from JSPS.
HS is thankful to the hospitality of Mathematics-for-Industry of Kyushu University
from October 22 of 2009 to January 7 of 2010, where part of this work has been
done.
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Index
absence of ground state
Nelson Hamiltonian, 199
Pauli-Fierz Hamiltonian, 105, 121
Agmon identity, 174
annihilation operator, 5, 8
Birman-Schwinger principle, 101, 104
Bogoliubov transformation, 20, 23, 38, 58
homogeneous case, 21
inhomogeneous case, 23
boson Fock space, 5
function space, 161
boson mass, 8
canonical commutation relation, 6, 10,
14, 31, 123
Chapman-Kolmogorov equality, 179
cocycle, 25
conjugate momentum, 9
contour integral, 45
coordinate process, 166, 174
creation operator, 5, 8
cut plane, 45
d’Alembertian operator, 151, 152
diagonalization, 37
ε self-energy term, 86
translation invariant, 36
ε self-energy term, 85
differential second quantization, 7
function space, 162
diffusion process, 174, 175
Dirac Hamiltonian, 29
Dirichlet form, 155
local, 155, 166
regular, 155, 166
Dirichlet principle, 194, 195
dispersion relation, 8, 32
variable mass, 149, 163
displacement operator, 23, 57
divergence form, 162
dressed electron state, 64, 65, 86
effective mass, 36, 71
running, 40
sharp cutoff, 40
sharp cutoff, 40
effective potential, 92, 127, 137
electric field, 33
enhanced binding, 99, 118
Nelson model, 129
Pauli-Fierz Hamiltonian
without scaling, 119
Pauli-Fierz model, 121
Euclidean scalar field, 182
Feynman-Kac formula, 127, 168, 183
finite particle subspace, 6
Fock vacuum, 5
function space, 161
free field Hamiltonian, 8
Nelson Hamiltonian, 123
variable mass, 164
Pauli-Fierz Hamiltonian, 32
ground state energy, 66, 71
204
asymptotics, 80
asymptotics for many particles, 83
many particle, 83
sharp cutoff, 80
harmonic oscillator, 73
Hilbert transform, 38
hydrogen-like atom, 29
IMS localization formula, 134, 141
infrared regular condition, 36
infrared singular condition, 36
intertwining operator, 21, 57
Jacobi coordinates, 139
Klein-Gordon equation, 150
pseudo Riemannian manifld, 150
Lamb shift, 30, 37, 92
Lieb-Thirring inequality, 154
local exponent, 26, 27
lowest two cluster threshold, 131, 132
momentum lattice approximation, 109
Nelson Hamiltonian, 124
variable coefficients, 149
variable mass, 164
Nelson’s analytic vector theorem, 10
number operator, 7, 36
partition of unity, 133
Pauli-Fierz Hamiltonian, 33, 34
translation invariant, 35
polarization vector, 31
positivity improving, 172, 184, 189
projective unitary representation, 25
pseudo Riemannian manifold, 150
quadratic operator, 15
radiation field, 32
reduced mass, 140
reflection symmetry, 174
Rollnik class, 106
scaling limit, 88
second quantization, 6, 8
function space, 162
Segal fields, 9
self-energy, 36
self-energy term, 84
shift invariance, 175
strong coupling limit, 91
super-exponential decay, 174
symplectic group, 13
one-parameter, 26
symplectic structure, 54
negative mass, 93
time evolution of radiation field, 63, 64
negative mass, 97
transversal delta function, 32
ultracontractivity, 168, 173
uniqueness of martingale problem, 168
unitary group
one-parameter , 28
UV cutoff, 79
sharp, 41, 79, 108, 121
variable mass, 149
von Neumann uniqueness theorem, 73
205
weak coupling limit, 89, 126, 127
Wick product, 10
function space, 162
206
